Tell me about the altitudes, medians, angle bisectors, and perpendicular bisectors of the sides of a triangle. Be sure to mention anything interesting that happens when the triangle is "special" - equilateral, isosceles, right, etc.
From: schwarzem@aol.com School: Beisenkamp-Gymnasium in Hamm, Germany SPECIAL LINES AND POINTS IN DIFFERENT TYPES OF TRIANGLES A hard work (the translation!) of students of the Beisenkamp- Gymnasium 1. ANGLE BISECTORS IN TRIANGLES Angle bisectors are lines which bisect the interior angles alpha, beta and gamma. Triangles with no special angles: The angle bisectors cut each other in one point (I). The angle bisectors cut in one point. That's why all points of an angle bisector have the same distance to the lines that form this angle. And if there is one point, the point I, that lies on two (three) angle bisectors it has the same distance to all lines that form the three angles alpha, beta, gamma. This point I has from all sides of the triangle the same distance. You can show this, if you construct the perpendiculars to the three lines (AB, BC, CA) through the point W. The distances are the same. The three sides are cut in one point which is called 'the point of contact' of the inscribed circle. The inscribed circle "cuts" all sides of the triangle only in one point so that the lines are tangents of the inscribed circle. Isosceles triangles: The base angles have the same value (60 degrees) and point C lies at the middle vertical of AB. So the point l I has the same distance from the edges A and B of the triangle. The length of the segment IC depends of the angle gamma. Is gamma bigger than 90 degrees the segment is smaller than IA and IB. This point (I) is the midpoint of the inscribed circle, which touches all lines (see above). Equilateral triangles: The point of intersection of the angle bisects is the point that lies in the middle of the triangle. The segments IA, IB and IC have the same length. I is the point of intersection of the Segment perpendicular bisectors of the sides. the angles AIB and AIC and BIC are 120 degrees. Rectangular triangles: One angle has the value 90 degrees. In every triangle the point I lies in the triangle. 2. BISECTORS OF THE SIDES Every triangle has a center of gravity, at which the three bisectors of the sides intersect. We call this point S. It lies in the triangle. The bisectors of the sides are constructed by connecting the point at the middle of one side with the edge of the triangle that it opposite to that side. The center of gravity divides every bisector of the sides so that the distance from the center of gravity to the edge is two times longer than the distance of the segment from S to the midpoint of the opposite side. In short: The center of gravity divides every bisector of the side in relation 2:1 . Proof: E,F are the centers of AC and BC, A* and B* the centers of AF and SB and S is the center of gravity . The triangle A*B*S and the triangle EFS are congruent because : 1) the lines EF are parallel to AB and the length of EF* =0.5 * length of AB (parallels in the middle of the triangle ABC) and the lines A*B* are parallel to AB and the length of A*B* =0.5 * length of AB (parallels in the middle of the triangle ABS). So: A*B* = EF and parallel. 2) SEF(angle) = A*B*S(angle) 3) B*A*S(angle) = SEF(angle)( Alternate angles on parallels ) The Congruence of the triangles is shown. We compare the following segments : ES = SB* = AE AS = AE + ES = 2 times B*S BS = BF + FS = 2 times A*S and so : CS = 2 times C*S We can explain why this point S is called the center of gravity: when you balance a triangle exactly at the center of gravity on a stick it is and remains in balance . 3. ALTITUDES IN A TRIANGLE What are altitudes in a triangle and how can we construct them? Altitudes are lines which are perpendicular of one side and go through the point of the triangle that is opposite to this side. In any triangle the altitudes have one point at intersection. We call it H. If one angle is bigger than 90 degrees, H lies outside the triangle ABC. In an equilateral triangle the altitudes correspond to the perpendicular bisectors of the side (and the angle bisectors and bisectors of the sides). In an isosceles triangle, all altitudes have one point of intersection too (also in acute-angled triangles). It divides the triangle in two congruent parts. It is one angle. In a rectangular triangle the point of intersection of all altitudes corresponds to the edge where we can find the angle of 90 degrees. In an obtuse-angled triangle the altitudes have one point of intersection outside the triangle (you see it when you construct the altitudes from the edges with the angles smaller than 90 degrees.) 4. PERPENDICULAR BISECTORS OF THE SIDES We made a table to compare many things which are interesting after having constructed the perpendicular lines in different types of triangles. The perpendicular lines of the sides have one point at intersection; we call it M. M has the same distance from all edges. (All points of one perpendicular line of one side of a triangle have the same distance to the two edges. And if there is a point M, that lies on two (or three) different lines it has the same distance to all points of the segments AB, AC, BC. M is the center of a circle. A, B, C lie on this circle. Here are the different types of triangles that we considered: 1. isosceles 2. equilateral 3. right angle and equilateral, too 4. right angle 5. obtuse angle 6. acute angle Point M 1. M Lies inside of the triangle; 2. inside of the triangle 3. at the center of the hypotenuse 4. at the center of the hypotenuse 6. outside of the triangle Angles around M that are formed by the perpendicular bisectors 1. all the same (120 degrees) 2. you can find two different angles around M 3. every of the 8 angles around M are 90:2 equal 360:8 equal 45 degrees 4. the angle of the perpendicular bisectors that form 90 degrees in the triangle is 90 degrees. 5. three different angles 6. three different angles Perpendicular bisectors that are altitudes 1. all three perpendicular bisectors of the sides are the altitudes, too 2. one perpendicular bisector of the sides lies on the altitude 3. one perpendicular bisector of the sides is identical to the altitude of this side 4. no perpendicular bisector of the sides is identical to the altitude of the side 5. see 4 6. see 4 Area of the part of the circumcircle in which we find the triangle ABC compared with the area of the semicircle with r equal to 0.5 * AB 1. bigger than the area of the semicircle 2. depend of the angle gamma, see 5, 6 3. same area as the semicircle 4. same area as the semicircle 5. bigger than the area of the semicircle 6. smaller than the area of the semicircle Perpendicular bisectors that are identical with bisectors of the angles 1. all perpendicular bisectors of the sides are identically with a bisector of the angles alpha, beta and gamma 2. one perpendicular bisector of the sides is identical with a bisector of an angle 3. see 2; it's the perpendicular bisector of the side opposite to the angle that has 90 degrees 4. all are different 5. all are different 6. all are different Bisectors of the sides that are identical to perpendicular bisectors 1) all; M = I = S = H 2) one 2) one 3) none 4) none 5) none 6) none
From: ruth@forum.swarthmore.edu (Ruth Carver) Celine McElwee, Amy Barbieri, and Sarah Schmalbach Grade 9 Mount Saint Joseph's Academy Triangles have many different properties which include altitudes, medians, angle bisectors, and perpendicular bisectors. Every triangle has three medians and three altitudes. An altitude of a triangle is the perpendicular segment from a vertex to the line that contains the opposite side. The three altitudes meet in a common point of intersection called the centroid. Please note that the altitude is not always contained within the triangle. The altitudes of an acute triangle are in the interior of the triangle. One altitude of a right triangle is inside, but the other two are the legs of the triangle. One altitude of an obtuse triangle is within the triangle, while the other two are on the exterior. A median of a triangle is a segment from a vertex to the midpoint of the opposite side. Therefore, the median bisects the opposite side. The intersection of the medians always occurs in the interior of a triangle. They also always lie in the same cross pattern with every type of triangle. The medians of a triangle intersect in a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. The medians of a triangle also intersect at the centroid of the the triangle. The center of a triangle is also referred to as the centroid. An angle bisector is a ray that divides an angle into two congruent adjacent angles. Every angle has a bisector. If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. Therefore, if a point is equidistant from the sides of an angle, you can determine that the point lies on the angle bisector. To construct an angle bisector without Geometer's Sketchpad, you must first draw a random angle. The next step would be to circle with the center being the vertex of the angle. The circle will intersect the angle at two points and you must label the points X and Y. The fourth step would be to use X and Y as the centers of two circles with the same radius as the original circle. The point at which the two circles intersect lies on the angle bisector. If you draw a line from that point to the vertex then you have the angle bisector. A perpendicular bisector of a segment is a line that is perpendicular to a segment at its midpoint. Using Geometer's Sketchpad is not the only way to determine the perpendicular bisector of a segment. You can do this by drawing a random segment AB. The next step would be to draw two circles with a radius greater than one half AB from A and B. When the two points of intersection (X and Y) of the two arcs are connected, you have found the perpendicular bisector of the segment. The point at which the perpendicular bisector intersects segment AB is Z. You then know that the two segments from X to A or B, or from Y to A or B are congruent. This is proved by SAS Theorem of the triangles AXZ and Triangle BXZ. Segment XA and XB form two adjacent triangles. You know from definition that AZ is congruent to BZ. And you also know that angle XZA and angle XZB are right angles. Lastly you know that XZ is congruent to XZ by reflexive property. Therefore, triangle AXB is congruent to triangle BXZ by SAS, and XA is congruent to XB because corresponding parts of congruent triangles are congruent. Isosceles triangles have certain special qualities. In an acute isosceles triangle, the segment from the vertex angle to the base is a median an altitude a perpendicular bisector and an angle bisector. In a right isosceles triangle the segment from the vertex to the hypotenuse is a median, altitude,perpendicular bisector, and an angle bisector. This segment also cuts off two acute isosceles triangles in which the segments form the vertex angles to the bases are medians, altitudes, perpendicular bisectors, and angles bisectors of each triangle. In an obtuse isosceles triangle, the segment from the longest side to the opposite angle is a median, altitude, perpendicular bisector and an angle bisector. Equilateral triangles also have certain special qualities. In an equilateral triangle the medians and the altitudes are the same line. Also, the medians and altitudes are perpendicular bisectors and angle bisectors. All of the segments meet at the centroid of the triangle.
From: ruth@forum.swarthmore.edu (Ruth Carver) Elissa Serrao and Lauren Wall Grade 9 Mount Saint Joseph Academy Thank you for choosing this moth's project to be about perpendicular bisectors, etc. Just recently, our class did a GSP paper dealing with these things. So I whipped out my trusty packet, and used it as a basis for my answer. I'll begin by defining the terms that I'll be using throughout my explanation. 1) perpendicular bisector - a line or ray that is perpendicular to the segment of its midpoint. 2) median - a segment from a vertex to the midpoint of its opposite side. 3) altitude - the perpendicular segment from a vertex to the line that contains the opposite side. 4) angle bisector - a line or segment that divides an angle in half. *Equilateral Triangles* The equilateral triangle had the most "special" characteristics of all the triangles. After constructing the medians, I discovered that they were all the altitudes of this triangle, because they each formed 90 degree angles. Even more interesting is that the medians/altitudes were also the perpendicular bisectors of each of the three segments of the triangle. I also found the medians were angle bisectors of the triangle's three acute angles. Thus, in an equilateral triangle, altitudes = medians = perpendicular bisectors = angle bisectors. *Isosceles Triangles* The altitudes of an isosceles triangle depend on whether the triangle is acute or obtuse. If the triangle is obtuse (which is the type I used) than only one of the altitudes is included in the interior of the triangle. The remaining two altitudes can be found by extending the lines of two of the sides of the triangle. If the triangle was acute, then all of the altitudes would be located in the triangle's interior. In the obtuse triangle that I drew, one of the medians was also the altitude, and the other two medians were congruent. The reason these two medians were congruent was because two of the sides of the triangle were congruent as well. *Right Triangles* Right triangles are unique because two of their altitudes are included in the triangle itself. (They are formed by the triangle's legs and its 90 degree angle.) None of the medians in a right triangle are congruent (unless it is an isosceles triangle), nor are the altitudes or medians perpendicular bisectors. *Scalene Triangles* Again, in scalene triangles, the number of altitudes on the triangle's interior depend on whether the triangle is acute or obtuse. (I used an obtuse triangle) The remaining two altitudes were found by extending the lines of the two sides of the triangle. None of this triangle's medians are congruent, because all the sides are of different lengths. *General Knowledge Found About all the Triangles* There were some things that I discovered both on GSP and in class recently, that applied to more than one of the triangles listed above. One is that all of the altitudes of the iscoceles triangles are angle bisectors. This is true because when the altitude of a triangle is dropped, the triangles can be proven congruent by the SAS postulate. (Or corresponding parts of congruent triangles are congruent) Another thing I learned in class that applied to all the triangles was that the all perpendicular bisectors of each meet in one point that is equidistant from all the vertices. This point can act as the center of a circle that is to be inscribed inside the triangle. When the medians of a triangle are constructed, than the point of intersection of all the medians acts as the center of the circle that is to be circumscribed about the triangle.
From: ruth@forum.swarthmore.edu (Ruth Carver) Kristy Giballa, Courtney Piper, and Joan Vivaldelli Grade 10 Mount St. Joseph Academy Sides of Triangles An altitude of a triangle is the perpendicular segment from a vertex to the line that contains the opposite side. With an acute triangle thee altitudes will always intersect inside the triangle. In the case of a right triangle two of the altitudes always create legs of the triangle. The third altitude always intersect the hypotenuse. In an obtuse triangle the altitudes always intersect each other outside of the triangle. The lines that contain the altitudes of a triangle intersect in a point. The median of a triangle is a segment from a vertex to a midpoint of the opposite side. In an isosceles, right, acute, obtuse, scalene, and equilateral triangle the medians always will intersect inside of the triangle. The median of an isosceles triangle, if from the vertex to the midpoint of the base side, is also an altitude and an angle bisector. The medians of a triangle intersect in a point that is 2/3 of the distance from each vertex to the midpoint of the opposite side. Angle bisector is the ray that divides the angle into two congruent, adjacent angles. The shortest distance from a point on the bisector of an angle to the rays, is always congruent segments perpendicular to the ray and passing through that point. The bisectors of the angles of a triangle intersect in a point that is equidistant from the three sides of the triangle. Once you draw these angle bisectors you can circumscribe a triangle around it. A perpendicular bisector is a line (ray or segment) that is perpendicular to the segment at its midpoint. The perpendicular bisectors of the sides of a triangle intersect in a point that is equidistant from the three vertices of the triangle. If a segment is cut by a perpendicular bisector and segments connect the endpoints to any point on the perpendicular bisector, the segments are congruent. If a point is equidistant from both sides of an angle then it is on the perpendicular bisector. The median of a triangle that will always be the altitude is the one which goes to the vertex angle to the midpoint of the base side. It can also be called a perpendicular bisector.
From: ruth@forum.swarthmore.edu (Ruth Carver) Lauren Grabowski and Sarah Joyce Grade 10 School: Mount St. Joseph Academy The altitude of a triangle is the perpendicular segment from a vertex to the line that contains the opposite side. In an obtuse triangle, two of the altitudes are outside of the triangle, and all three of the altitudes intersect outside of the triangle. In an acute triangle, the altitudes intersect inside of the triangle. In an isosceles triangle, from the vertex angle, the altitudes are perpendicular bisectors and medians. This of course holds true for any angle in an equilateral triangle. In a right triangle, two of the altitudes act as legs, while the third altitude lies inside the triangle. The altitude that lies in the right triangle is the geometric mean between the segments in the hypotenuse. Also, in a right triangle, all three altitudes intersect at the angle. The median of a triangle is a segment from a vertex of a triangle to the midpoint of the opposite side. In a scalene triangle, all the medians always intersect at one point. For an equilateral triangle, all the medians are congruent. In an isosceles triangle, the medians of the base angles are congruent and the median of the vertex is an angle bisector. An angle bisector is the ray that dived an angle into two congruent, adjacent angles. In an isosceles triangle, when there is an angle bisector on the vertex, two congruent right triangles are formed. In any triangle, by constructing a perpendicular line from any two intersecting angle bisectors, and forming a circle at the point of intersection to where the angle bisectors meet the triangle's sides forms a circle iscribed in a triangle. Finally, the perpendicular bisector is a line or segment that is perpendicular to the segment at its midpoint. Like an angle bisector, in an isosceles triangle, when there is a perpendicular bisector on the base side, two congruent right triangles are formed. Also, the point of intersection of two perpendicular bisectors of any triangle, is the center of the circle that is circumscribed about the polygon.
From: ruth@forum.swarthmore.edu (Ruth Carver) Jen Keeney, Kathleen Wuerth, and Omua Ahonkai Grades 10 & 9 School: Mt. St. Joesph Academy There were many aspects of triangles we were asked to explain this month. 1. medians: A median is a segment from a vertex of a triangle to its opposite side. 2. angle bisector: An angle bisector is a ray that divides an angle into two congruent angles. 3. perpendicular bisector: A perpendicular bisector is a line/ray/segment that is perpendicular to a segment at its midpoint. 4. altitude: An altitude is the perpendicular segment from a vertex to the line that contains the opposite sides. Medians cut their opposite sides into 2 congruent segments. In other words, if a median intersected line ABC at point B, AB is congruent to BC. When all 3 medians of a triangle are constructed, their point of intersection lies 2/3 of the distance from each vertex to the midpoint of the opposite side. Angle bisectors cut the angle into 2 congruent angles. If a point lies on an angle bisector, that point is equidistant from the sides of the angle. The bisectors of the angles of a triangle meet in a point equidistant from the 3 sides of the triangle. The bisector of the vertex angle of an isoseles triangle forms 2 congruent triangles. The bisector of any angle in an equilateral triangle also forms 2 congruent triangles. If a point lies on the perpendicular bisector of a segment, it is equidistant from that segment's midpoints. If 2 segments are drawn from a point on a perpendicular bisector to the endpoints, 2 congruent triangles are consructed. The perpendicular bisectors of the 3 sides of a triangle meet in a point equidistant from the 3 vertices. An acute triangle has 3 Altitudes within itself - a right triangle has 1 within itself and the other 2 are the legs, and an obtuse triangle has 1 within itself and 2 outside. In a right triangle, the length of the altitude to the hypotenuse is the geometric mean between the 2 segments of the hypotenuse. When the same altitude is drawn, each leg of the triangle is the geometric mean between the entire hypoteneuese and the segment of the hypoteneuse that they are adjacent to. Another feature of right triangles is that when the altitude to the hypoteneuse is drawn, the 2 triangles formed are similar to each other and to the original triangle.
From: ruth@forum.swarthmore.edu (Ruth Carver) Melissa DiFeo Grade 9 School: Mount St. Joseph Academy When I drew the medians of an isosceles triangle , they intersect at the centroid. When I drew the medians of a right triangle they also intersect at the centroid, which is equidistant from each of the three vertices. From the vertex angle of an isosceles triangle,not from the base angles, the median, the altitude, the angle bisector, and the perpendicular bisector are all the same segment because it forms a 90 degree angle, it bisects the angle, and it bisects the opposite leg. In a right triangle, 2 of the altitudes are already there because the 2 legs form them. The only altitude you have to construct is the one from the 90 degree angle to the hypotenuse. If you construct 2 perpendicular bisectors from the 2 legs of the right triangle, then it would form a rectangle from the 90 degree angle. When you form all the perpendicular bisectors, they all intersect in one point which is the midpoint of the hypotenuse, which is equidistant from each of the three vertices. When you form the angle bisectors of the triangle they intersect in exactly one point. When you construct the medians of an equilateral triangle you also form the angle bisectors because they cut the angles into 2 congruent angles, they also form the perpendicular bisectors because they cut the opposite angles into 2 congruent segments and also form a 90 degree angle; they also form the altitudes because they form 90 angles. Each separate median, angle bisector, perpendicular bisector, and altitude cut the equilateral triangle into 2 similar and congruent triangles. When you do it from each angle, they all intersect in exactly one point, which is equidistant from each vertex.
From: ruth@forum.swarthmore.edu (Ruth Carver) Jenn Cody and Rosie Twomey Grade 10 School: Mt. St. Joseph Academy The median of a triangle is a segment from a vertex to the midpoint of the opposite side. In all triangles, scalene, obtuse, acute, and right, the medians are always inside the triangle, and they always intersect at the same point and that point is the center of gravity. The altitude of a triangle is the perpendicular segment from a vertex to the line that contains the opposite side. Acute: 3 altitudes all inside triangle Right: 2 altitudes are the legs and third is inside Obtuse: 2 altitudes are outside and third inside The perpendicular bisector of a segment is a line (or ray or segment) that is perpendicular at its midpoint. Any point on a perpendicular bisector is equidistant from the endpoints of the segment. The perpendicular bisectors of the sides of a triangle also intersect in a point that is equidistant from the three vertices of the triangle. This allows you to circumscribe a circle about the triangle. Any point on the bisector of an angle will always be equidistant from the sides of the angle. If the angle bisector of the vertex of an isosceles triangle is constructed then it will also be a perpendicular bisector. The bisectors of a triangle also intersect in a point that is equidistant from the three sides of the triangle. This allows you to circumscribe a circle in the triangle. If the median is drawn from the vertex of an isosceles triangle or any angle of an equilateral triangle, it will also be its altitude making it its perpendicular bisector.
From: ruth@forum.swarthmore.edu (Ruth Carver) Jill Sommer Grade 10 School: Mt. St. Jospeh Academy Medians - - Since medians are segments from a vertex angle to the midpoint of the opposite side, the medians of every triangle are contained within the triangle. The medians of a triangle will meet at a point that is 2/3 of the distance from each vertex to the midpoint of the opposite side. Altitudes - - Altitudes are perpendicular segments from a vertex to the line that contains the opposite side. In a right triangle, 2 of the altitudes are the legs of the triangle. The other altitude bisects the hypotenuse of the triangle. In an acute triangle, all 3 altitudes are contained inside the triangle. In an obtuse triangle, 2 of the altitudes are outside the triangle. The lines that contain the altitudes will intersect in one point. Perpendicular Bisector - - A perpendicular bisector of a segment is the line (or ray or segment) that is perpendicular to the segment at its midpoint. Any point on this line will be equidistant to the endpoints of the segment. The perpendicular bisectors of the sides of a triangle meet at a point that is equidistant from the vertices. Angle Bisectors - - An angle bisector is a ray that divides an angle into two, congruent, adjacent angles. Any point on an angle bisector is equidistant from the sides of the angles. The angle bisectors of a triangle meet in a point that is equidistant from the sides of the triangle. Sometimes, these terms are interchangeable in triangles. For example, a median in an isosceles triangle is its perpendicular bisector, altitude, and angle bisector.
From: ruth@forum.swarthmore.edu (Ruth Carver) Lauren Goldbeck and Lindsay Pio Grade 9 School: Mount St. Joseph Academy There are many characteristics of triangles. For example, using the midpoints as vertices and conecting them you can make a similar triangle that is inscribed in the original. The altitudes of a triangle run from the angle to the opposite side and intersect the line at a 90 degree angle. An altitude must always stay perpendicular to the opposite side even if triangle is dragged into different shapes. In a right triangle, two of the altitudes are parts of the triangle. The third altitude is inside the triangle. In an acute triangle. the three altitudes are all inside the triangle. In an obtuse triangle, two of the altitudes are outside the triangle. A median runs from the angle to the opposite side and bisects that segment. Unlike the altitude, the median does not have to intersect the opposite side at a 90 degree angle. In a right triangle, the point of intersection of the three medians is the center of gravity. The medians of a triangle intersect in a point that is equidistant from the three vertices of the triangle. A perpendicular bisector of a segment is a line that is perpendicular to the segment at its midpoint. The perpendicular bisector is used in these theorems: 1. If a point lies on the perpendicular bisector of a segment then the point is equidistant from the endpoints of the segment. 2. If a point is equidistant from the endpoints of a segment then the point lies on the perpendicular bisector of the segment. Perpendicular bisectors of any type of triangle can be used to circumscribe a circle around the triangle. Also, using perpendicular bisectors of any triangle you can inscribe a circle at the point of intersection to inscribe it. The perpendicular bisectors of the sides of a triangle intersect in a point that is equidistant from the three vertices of the triangle. The bisector of an angle is the ray that divides the angle into two congruent, adjacent angles. There are several theorems that involve angle bisectors: 1. If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. 2. If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. As you can see, there are many interesting facts on triangles and their altitudes, medians, perpendicular bisectors, and their angle bisectors. By exploring these different aspects of triangles many things can be found that helps you to understand geometry better.
From: anne_sandler at shhs1.ccsd.k12.co.us Kelly Van Husen and Susie Sandstede Grade: 9 School: Smoky Hill High School The altitudes of any triangle form right angles and sometimes bisect the opposite side of the triangle. The altitude goes from the vertex and forms a line perpendicular to the opposite side of the triangle. In an equilateral triangle, the altitude bisects the opposite side, so it also becomes the median. When there is an altitude in an equilateral triangle, the altitude forms two congruent right triangles. The length of the altitude is x square roots of 3, the leg of the right triangle is x, and the length of the hypotenuse is 2x. In an isosceles triangle, the altitude also forms two congruent right triangles, and the lengths of their sides are the same as that of the equilateral triangle (x, x square roots of 3, and 2x). In a right triangle, the altitudes divide the triangle into similar right triangles. The length of the altitude is the square root of the product of the base legs of the two similar triangles. There are other formulas that can be used to find the lengths of other parts of a triangle like this. A median is drawn from the vertex of an angle to the midpoint of the opposite side of a triangle. A median can also be an altitude, but it doesn't have to be, and a median always bisect the segment it is drawn to. In an equilateral triangle, the medians, altitudes, and perpendicular bisectors are all the same lines, and therefore they are congruent. In an isosceles triangle, the median from the vertex angle always divides the triangle into two congruent triangles. The median from the vertex angle is always an altitude. In a right triangle, if a median is drawn form the vertex of the 90 degree angle bisects the angle and the opposite side and is also an altitude. The median divides the triangle into two congruent right triangles. In an isosceles right triangle, the median from the vertex of the 90 degree angle is also the altitude. The medians from the vertices of the base angles bisect each other. The perpendicular bisectors of a triangle are drawn from the vertex to the opposite side where they form a 90 degree angle and bisect the side - sort of like a combination of the altitude and median. The same principles apply to a perpendicular bisector of a triangle that they do when the altitude and median of any triangle are the same. An angle bisector of a triangle can also be the altitude, median, or perpendicular bisector of the triangle. It divides the angle into two congruent angles. In an equilateral triangle, the angle bisectors are all congruent and the same limes as the medians, altitudes, and perpendicular bisectors of the triangle. In an isosceles triangle, the angle bisectors of the base angles are congruent and bisect each other. The angle bisector of the vertex angle is also the median, etc. In a right triangle, the angle bisector of the right angle forms two 45 degree angles, divides the triangle into two similar triangles, and is the same as the median, etc. In an isosceles right triangle, the principles of an isosceles triangle and a right triangle apply, and the angle bisectors of the base angles divide the angles into two congruent 22.5 degree angles.
From: mark_overton@shhs1.ccsd.k12.co.us Mark Overton, Scott Bridger, Bryon Joel Grade: 9 School: Smoky Hill High School An altitude goes from a vertex to the opposite side and forms a right angle with it. If the triangle is equilateral, then the altitude will bisect the angle and bisect the opposite side and form 2 congruent triangles. If the triangle is isosceles, then the altitude will bisect the vertex where the legs meet and the opposite side as well as form 2 congruent triangles. The only altitude that can be drawn from a right triangle is from the right angle and will bisect the angle and the opposite side and also form 2 congruent triangles. Medians are drawn from a vertex to the midpoint of the opposite side. In an equilateral triangle the median will bisect any angle and from 2 congruent triangles. In an isosceles triangle, a median from the center vertex will divide the triangle into 2 congruent triangles. A median from the right angle of a right triangle will also create two congruent triangles. All triangles formed are right triangles. In an equilateral triangle the angle bisector is also the median and the altitude. In an isosceles triangle, the angle bisector from the center vertex is also the median and altitude In a right triangle, the angle bisector from the right angle is again the median and altitude. In an equilateral triangle the perpendicular bisector of any side is also the median and the altitude. In an isosceles triangle, the perpendicular bisector of the center vertex is the median and altitude, but if drawn to any other side you create a smaller version of the same isosceles triangle. A perpendicular bisector of the right angle of a right triangle is also the median and altitude. If drawn to any other side then you again form a smaller version of the same triangle.
From: anne-d.-sandler@shhs1.ccsd.k12.co.us Somsnit Vanprapa Grade: 10 School: Smoky Hill High School An altitude of a triangle is a line segment drawn from any vertex of the triangle to the opposite side, extended if necessary, and perpendicular to that side. (An altitude of a triangle forms right (90) an angles with one of the sides.) If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the given right triangle and to each other, the altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse, either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg. A median of a triangle is a line segment drawn from any vertex of the triangle to the midpoint of the opposite side. (A median divides into two congruent segments, or bisects the side to which it is drawn.) When the triangle is equilateral, all medians are congruent. When the triangle is isosceles, two medians from the base angles are congruent. When the triangle is a right triangle, no medians are congruent. When the triangle is obtuse, no medians are congruent. An angle bisector is a ray that bisects an angle of a triangle, it divides that opposite side into segments that are proportional to the adjacent sides. When the triangle is isosceles, the angle bisector from the vertex (not the base angles) bisects the segment. When the triangle is equilateral, the angle bisectors bisect all of the segments it passes through. When the triangle is a right triangle, no angle bisector bisects the segment. The perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment.
From: ssafavi@vaultbbs.com Sean Mostafavi Grade: 10 School: Smoky Hill High School 1. The angles in a triangle always add up to 180 degrees. 2. The area is the 0.5(b*h) where base is the length of a side and h is the perpendicular distance from that side to the angle opposite it. 3. Angles can be bisected by doing the following - pick a corner - mark a spot equal distance from that angle on its two adjoining sides - use a compass to make an arc from each of these two points - draw a line from the corner through the intersection of the two arcs and the angle is now bisected. 4. An equilateral triangle has equal sides and equal angles of 60 degrees. The altitude from any angle bisects the opposite side. 5. An isosceles triangle's two base angles are equal in degrees. The altitude of the third angle bisects the base.
From: LIMBERJ@mail.firn.edu Jaime Uhazie Grade 9 Martin County High School, Stuart, FL To start this problem, first I will define each of these terms: altitude, median, angle bisector, and perpendicular bisector. I will also state any theorems about the terms. ALTITUDES: The perpendicular segment from a vertex to the line containing the opposite side. In an acute triangle, the three altitudes are all inside the triangle. In a right triangle, two of the altitudes are parts of the triangle. They are the legs of the triangle. The third altitude is inside the triangle. In an obtuse triangle, two of the altitudes are outside the triangle. When the triangle is isosceles, the altitude is a line from the vertex to the midpoint of the opposite side. When the triangle is right isosceles, two of the altitudes are the sides of the triangle, and the third altitude divides the hypotenuse into two equal segments. MEDIAN: A segment that forms a vertex to the midpoint of the opposite side. The median divides the opposite segment into congruent segments. In an isosceles triangle, the median divides the triangle into two congruent right triangles. In a right isosceles triangle, the median also serves as an altitude. ANGLE BISECTOR: The ray that divides the angle into two congruent adjacent angles. Angle bisector theorem: If BX is the bisector of <ABC, then m<ABX=1/2 m<ABC, and m<XBC=1/2 m<ABC. This means that when an angle is bisected, one part of the angle is half as large as the whole angle. In isosceles triangles, the angle bisector is also the median. In a right isosceles triangle, the angle bisector is also the median and the altitude. In an equilateral triangle, the angle bisector also serves as the median. Theorems: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. If a point is equidistant from the sides of an angle, then, the point lies on the bisector. PERPENDICULAR BISECTOR: A line, ray, or segment that is perpendicular to the segment at it's midpoint. In a given plane, there is exactly one line perpendicular to a segment at its midpoint. Theorems: If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. This means that the ends of the segment are congruent if placed anywhere on the perpendicular bisector. If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. In an obtuse triangle, the perpendicular bisector is not in the triangle.
From: LIMBERJ@mail.firn.edu Laura Ejups, grade 10 Sara Holtzman, grade 9 Martin County High School, Stuart, Florida An altitude of a triangle is the perpendicular segment from a vertex to the line that contains the opposite side. In an acute triangle, the 3 altitudes are all inside the triangle. In a right triangle, 2 of the altitudes are parts of the triangle. They are the legs of the right triangle; the third is inside. In an obtuse triangle, 2 of the altitudes are outside the triangle; the third is inside. In equilateral, scalene, and iscoceles triangles, all of the altitudes are inside the triangle. A median of a triangle is a segment from a vertex to the midpoint of the opposite side. In right, acute and obtuse triangles, the medians are all contained inside the triangle. In an equilateral triangle, each median is the same distance apart from each of the sides (they are all the same). In an isosceles triangle, two of the medians appear the same, and in a scalene triangle, all medians are different and located inside the triangle. An angle bisector is the ray that divides the angle into 2 congruent adjacent angles. In every type of triangle, the angle bisector is located inside of the triangle. In an equilateral triangle, the angle bisector meets the opposite side at the same point for each of the 3 sides. For an iscoceles triangle, 2 of the angle bisectors, (the ones that meet at the congruent sides), meet the opposite sides at the same point as the others. A perpendicular bisector of a segment is a line, (or ray or segment), that is perpendicular to the segment at its midpoint. The perpendicular bisector is at a different point in each of the isoceles, scalene, right, obtuse, and acute angles. It is at the same point in the equilateral triangle.
From: mlmann@stgeorges.edu Win Kelly and Mike Mann Grade: 10 School: St. Georges School Altitudes - The altitude is drawn from the vertex of a triangle to the opposite side, meeting it at a right angle. The point where the three altitudes of a triangle meet is called the orthocenter. Angle bisector - Line cutting the vertex angle in half, the point where the three angle bisectors intersect is called the incenter, which is the center of the inscribed circle. Medians - The median is a line from a vertex of a triangle and intersects the opposite side at its midpoint. The point at which the three medians intersect is called the centroid. This point divides all three medians into two segments, of 1/3, 2/3 ratio. Perpendicular bisector - This segment is a line from the midpoint of a side drawn at a right angle. The point where all the perpendicular bisectors meet is called the circumcenter.
From: mailee_eskelund@stgeorges.edu Grade: 10 School: St. Georges School Altitudes are segments that go from one angle to the opposite side perpendicularly in any triangle. Medians are segments that go from one angle to bisect the opposite side. Angle bisectors are the rays that bisect angles, making both angles congruent to each other. Perpendicular bisectors are segments or lines that bisect other sides or each other.
From: jacobcrowell@stgeorges.edu Jacob Crowell Grade: 9 School: St. Georges School Answer: An altitude runs from any vertex angle to the opposite side and is perpendicular to that side. A median runs from any vertex angle and cuts the side opposite to this angle into two congruent halves. An angle bisector runs from any vertex angle to the opposite side and cuts the vertex from which it runs into two congruent angles. A perpendicular bisector runs from any point on the perimeter of the triangle to another side creating two right angles and cutting the segment into two congruent halves. When a triangle is isosceles or equilateral, the median, altitude, perpendicular bisector and angle bisector are the same line.
From: ELDK81A@forum.swarthmore.edu Stephanie Balster Grade: 8 School: Thomas Russell Middle School (Milpitas, CA) The altitudes of a triangle intersect in a point. If it is an obtuse triangle, they intersect outside of the triangle. If it is a right triangle, they intersect on the vertex of the right angle. The medians of a triangle intersect in a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. The angle bisectors of a triangle intersect in a point that is equidistant from the three sides of the triangle. The perpendicular bisectors of a triangle intersect in a point that is equidistant from the three vertices of the triangle. In a obtuse triangle, they intersect outside of the triangle. In a right triangle, they intersect on the hypotenuse of the triangle.
7 December 1996