Construct an Isosceles Triangle - April 15-19, 1996
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Give me at least three different ways to construct an isosceles triangle.Now, "construct" does not mean "draw two lines that are the same length and connect them with a third." That is not a construction. A construction is something that you could repeat over and over and always get an isosceles triangle.
Answers that include more than three solutions are more than welcome.
Annie says: Solutions
Twenty-five correct solutions this week, and none wrong (several people initially submitted incomplete answers, but everyone corrected them).
I really like this problem! I often use it in introductory Sketchpad workshops for teachers. But I had ulterior motives for using it this particular week. Key Curriculum Press (publisher of Sketchpad) asked me to man a table at their Sketchpad Users' Group meeting at the NCTM National Conference in San Diego. I suggested that I show how students have been using Sketchpad to solve the Problem of the Week.
I decided to post this question in hopes that some of you would send in Sketchpad-based solutions, and you did! The meeting was a success, and teachers were really impressed with the work that students have been doing.
Some of the solutions that I've accepted aren't exactly correct - while they do create isosceles triangles, they create ones with restrictions - only 60 or 90 degree angles, for example. A true construction will be 'general'; that is, you could get any isosceles triangle out of it, with any length sides and any size angles. So, for example, the isosceles triangle constructed from two radii of a circle is good, because you can change everything - but the isosceles triangle made by drawing the diagonal of a square isn't good, because you are stuck with a 90 degree angle.
So whenever someone says "construct," try to be as general as possible, with no restrictions on the final figure beyond those given in the problem.
Following are highlights. The names of all the people who submitted correct solutions and most of the solutions are also available, and we welcome one new school this week.
Brooke Freeman Grade: 10 School: Livermore High 1) Use a compass to draw a circle with any radius. Draw any diameter and use arcs to find the perpendicular bisector of that segment. Continue the line until it reaches the circle. With those three points draw the other two lines to that point. 2) Use a compass and draw a circle, draw any chord that you please, bisect that chord, and connect the remaining two chords. 3) Use a compass and draw a circle, draw two perpendicular diameters in the circle, use those four points to draw a square, then draw either of the diagonals of that square.
Mike Sue, Neil Tucker, Deanne Derego Grade: 10 School: Granada One way: Draw a triangle with 2 congruent adjacent angles. That makes the opposite legs congruent, which means it can be an isosceles triangle. Second: Draw a circle. Draw one chord in the circle but it can't be the diameter. Next connect the two points with a segment where the chord meets with the circle and the middle of the circle. This will give you two radii, and they obviously will both be congruent. Third: Draw a line segment of any size. Then draw an arc from one end of the segment and on the other end, another arc that is congruent to the first. Find where the intersection is and draw line segments from the segment and the point of intersection. This will give you an isosceles triangle as well as the others. [They sent another one at my prompting - see if you can decide which three are best. - Annie] In response to your response, draw line segment AB. At each endpoint draw an arc. Where the arc intersects the segment draw another so it intersects the previous arc. From each end point draw a line to its arc intersecting at point C. The segment and these form an isosceles triangle, triangle ABC.
Amy Forster Grade 7, age 11 Wilkins/Forster family Crooked Tree Point Cygnet, Tasmania, Australia I have found 6 ways to construct an isosceles triangle. Construction 1 1. From one point draw 2 straight lines the same length. 2. Connect the ends of the lines, where they are furthest apart, with a third straight line. Construction 2 1. From a single point draw two straight lines any angle apart. 2. Place a pair of compasses, set at any length, on the point where the lines meet and draw an arc which intersects both lines. 3. Draw a third straight line which joins the two points where the arc intersects the lines. Construction 3 1. With a pair of compasses draw a large arc of a circle. 2. From the centre of the circle draw 2 lines which meet the arc at any 2 points. 3. Draw a third line which joins the 2 points lying on the arc. This method will give you a range of isosceles triangles, each with 2 sides equal to the radius of the arc. Construction 4 1. Draw a circle and draw in a diameter. 2. At any point on the diameter, draw a line at right angles to the diameter which intersects the circumference of the circle on each side of the diameter, at points A and B. 3. Choose one of the 2 points where the diameter intersects the circle and from it draw 2 lines, one to point A, and one to point B. Construction 5 1. Draw a line of any length. 2. Set a compass at any length greater than half the length of the line just drawn. 3. Set the point of the compass at one end of the line and draw an arc. Do the same at the other end of the line. From the point where the two arcs intersect draw 2 straight lines, 1 to each end of the original line. Construction 6 (This is a more practical solution so I'm not sure if it is what you want.) 1. Take any rectangular piece of paper. 2. Cut it in half along one of its diagonal lines. 3. You now have two right angled triangles. Place the two second longest sides together along their length(or the 2 short sides together) so that one triangle is a reflection of the other (i.e., the right angles are next to each other). Together they form an isosceles triangle. ************************************************ Thomas S. Kuo Grade: 7 School: Murray Junior High School, Ridgecrest, California * * A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * B * D C * Method 1: (1) Draw a line segment BC. (2) Expand compass to the length of legs of the isosceles triangle. (3) Put needle of the compass at point B and draw a circle. Put needle of the compass at point C and draw a circle. These two circles interact at point A. (4) Connect points A, B, and C. An isosceles triangle with sides given is built. [Thomas added some explanation later - do you think it helps?] If the lengths of two congruent sides and the third side of the isosceles triangle are given, then the length of BC should be equal to the length of the third side and the radius of circle should be equal to the length of the two congruent sides. Then it should be clear. The radius of circle should be greater than half of the length of BC or they can not form a triangle. Method 2: (1) Draw two line segments AD and BC. They are perpendicular to each other and interacts at point D. (2) Make BD = DC = half of length of side of isosceles triangle. (3) Expand compass to the length of legs of the isosceles triangle. (4) Put needle of the compass at point B and draw a circle. The circle and line AD interact at point A. (5) Connect points A, B, and C. An isosceles triangle with sides given is built. [more added later:] Again, let lengths of two congruent sides and the third side of the isosceles triangle be given. Steps (2) and (3) above should be rewritten as follows: (2) Make BD = DC = half of length of the third side of the triangle. (3) Expand compass to the length of the two congruent legs of the triangle. In method 1, point A is determined by the interaction of two circles. In method 2, point A is determined by the interaction of one circle and the bisection line of line segment BC. Method 3: (1) Draw a circle centered at point A with radius the length of legs of the isosceles triangle. (2) Find any point B on the circle and connect point A and B. (3) Draw a circle centered at point B with radius the length of side of the isosceles triangle (other than the equal legs). This circle will interact with the previous circle at point C. (4) Connect points A, B, and C. An isosceles triangle with sides given is built. [I asked Thomas if he could make this more general] Let the lengths of two congruent legs and the third leg of the isosceles triangle be given again. The step (1) and (3) above should be rewritten as follows: (1) Draw a circle centered at point A with radius as the length of the third leg of the triangle. (3) Draw a circle centered at point B with radius as the length of the two congruent legs of the triangle. This circle will interact with the previous circle drawn in (1) at point C. The difference between this method and the previous methods is that I draw the congruent leg first. All methods above are assumed that lengths of legs of the isosceles triangle are given. However, as stated in method 1, as long as the length of the congruent legs are greater than half of the length of the third leg, these methods should work.
Katie Walder and Lindsay Parsons Grade 9 School: Mt. St. Joseph Academy, Flourtown, PA
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