A Math Forum Project

Geometry Forum - Problem of the Week Solutions - Building a Bridge Across a Chasm, March 20-24, 1995 Annie says: Seems a fair number of people just happened to have done congruent triangles! A lot of correct answers this week. What's interesting is that so many answers could all be right but be so different - not in how the answer was found, but in how it was explained. Reading the solutions may give many of you a good idea of how to write shorter answers - long answers aren't bad, but they can sometimes be confusing. And when solutions to problems require longer answers, it pays to know how to be as brief as possible. One technique used by many of the respondents is to first label the points, then use the labels in the explanations. This cuts down on a lot of words, and makes thing easier to follow. Jessica Muniz Jessica Muniz, grade 9, Fairfield HS The edge of the chasm is directly across from the tree, so the trail formed when they walk 20 paces along the chasm is perpendicular to the log or rope that will extend from the chasm to the tree. Therefore, a 90 degree angle is formed with the vertex at the chasm. Then when he walks 20 paces farther from her, this equals her length from the edge of the chasm to her present stance. Then when he turns a 90 degree angle this angle is congruent to the angle with its vertex at the chasm. If an imaginary line was formed at the time she was directly in the line of sight between him and her, two right triangles have been formed. The vertical angles in the triangles, made by the imaginary line and trail paced, are congruent. Therefore, by LA (leg-angle), the triangles are congruent and by CPCTC (corresponding parts of congruent triangles are congruent), the distance he walked from the chasm is congruent to the distance from the chasm to the tree. Julie Black Julie Black, grade 9, Fairfield HS The "measuring thing" worked because after every step, you ended up with two congruent triangles. After drawing the picture, you had two triangles. Both of them were right triangles because each triangle had a leg that made a right angle with an edge of the chasm. The triangle's other legs were congruent because they were each 20 paces. Then, I realized that the angles closest to those two congruent legs were also congruent because they were vertical angles. Therefore, the triangles are congruent because of LA (leg-angle) and all of the parts of the triangle are congruent. So the other legs are also congruent. Samantha Brenner Samantha Brenner, grade 9, Fairfield HS The line from the start to the tree is parallel to where the guy turns 90 degrees because they are each parallel to the same line in a plane. The picture forms two congruent triangles: congruent right angles, corresponding sides that are each 20, and a pair of congruent alternate interior angles. The triangles are congruent because of AAS. The distance that Bill walked from the chasm is congruent to the distance across the chasm because corresponding parts of congruent triangles are congruent. 2nd Period Geometry Class Ridgeview High School Bakersfield, California Geometry Class - 2nd period Annie, Here is our solution. To begin with, we drew a line segment 20 paces long from the point across from the tree. We placed point for the girl and drew another line segment of 20 paces. We drew a line segment at a 90 degree angle until the girl and tree lined up. We noticed that we had two right triangles. We know that the angles formed at the girl point are vertical angles so they are congruent. The line segments are congruent because they are both 20 paces. We also have two right angles because they are both right triangles. Therefore, the triangles are congruent by the Angle Side Angle Theorem. Furthermore, congruent triangles implies congruent parts. So, the distance across the chasm is equal to the distance he walked back from the chasm. 1st Period Geometry Class Ridgeview High School Bakersfield, California Geometry Class - 1st period Annie, Here is our solution. By his walking, he creates two right triangles. The triangle across the chasm we will name Triangle TDG with the tree at point T, the girl at point G, and the point directly across from the tree at point D. The other triangle we will label Triangle ABG with the man at point A and the his turn of 90 degrees at point B. Angle DGT and Angle BGA are congruent because of vertical angles. Angle D is 90 degrees because it is directly across the chasm from tree. Angle GDT and Angle GBA are congruent because they are both 90 degrees. Side BG and Side DG are congruent because they are both 20 paces. Therefore, Triangle TDG is congruent to Triangle ABG because of the Angle Side Angle theorem. Therefore, TD and AB are congruent because congruent triangles implies congruent parts. 7th Period Geometry Class Ridgeview High School Bakersfield, California Geometry Class - 7th period Annie, Here is our solution. We can form two right triangles from his walking. Point A is directly across from the tree. Point B is the tree. Point C is where the girl is standing. Point D is where he turned and Point E is where he ended up. Length AC is congruent to Length CD because they are both 20 paces. Angle ECD is congruent to Angle ACB because they are vertical angles. Angle BAC is congruent to Angle CDE because they are both right angles. Therefore, the two triangles are congruent because of the Angle Side Angle Theorem. Since congruent triangles have congruent parts, the distance he walked is equal to the distance across the chasm. Adam D. Shapiro My name is Adam D. Shapiro and I'm in the 10th grade at Akiba Hebrew Academy, in Merion, PA. The name of my Geometry teacher is Mrs. Perez. The first step I took was to connect a line from where the tree is on one side of the chasm, directly to the other side. I then made a line 20 paces to the right, and marked it, then another 20 paces to the right and marked it, I made a 90 degree turn and went down far enough so that I could draw a line directly through the point I marked (at the first 20 paces) and then to the tree. What I have now created at this point are two triangles. Once I had the two triangles I went on to prove their congruency. First, the two sides that represent the 20 paces they each walked are congruent, therefore I have one set of sides congruent. Second, the two triangles together have vertical angles and by the theorem they are congruent and they give me a set of congruent angles. Before we do the third step I'd like to make it clear that I named the line from the tree to the other side of the chasm-- x, and the line that was made after he made his 90 degree turn I labeled--- y. Line x is drawn from the tree directly to the other side therefore forming a right angle (90 degrees) because it is the most direct and shortest way across, the other right angle (90 degrees) can be found in the other triangle at the point that the boy made a 90 degree turn, this information gives me my second and final set of congruent angles. With all this information I have give, you can prove these triangles congruent by the ASA Postulate (Angle Side Angle) Since the two triangles are congruent all sides are congruent because, corresponding parts of congruent triangles are congruent. This is also saying that x = y and the distance from the tree to the other side equaled the distance he walked after making the 90 degree turn. Thank you, Adam Shapiro E-mail: scraps9636@aol.com Eli Segal Eli Segal Akiba Hebrew Academy 10th Grade One can tell that he is correct by creating two triangles. Call the point of the tree A and the length from the tree to the other side AB. Also call the point where the girl stops C and the point where the boy turns is D, while the point where the boy stops is E. Thus triangles ABC and EDC are formed. We already know two sides to be congruent, BC and CD because they both equal 20. Angle ACB and angle ECD are congruent by the vertical theorem. The measure of angle ABC equals 90 because the shortest distance from a point to a line is always perpendicular to that line. In the given information, the measure of angle CDE also equals 90. Therefore, by the ASA congruency theorem, triangle ABC and triangle EDC are congruent. Therefore AB (distance across the chasm) and DE (the distance walked by the boy after his turn) are equal cause corresponding part of congruent triangles are congruent. Katie Kaminski and Janine Peterson I hope it's not too late, but two of my students solved the problem of the week for March 20-24. The tree is point D, the marked spot is E, the girl stands at C, the boy walks to A, then turns to point B. the tow resulting triangles are congruent by ASA: The two right angles are congruent, the vertical angles are congruent, and the two 20 pace sides are congruent. The students concluded that AB = DE by CPCTC. Their names are Katie Kaminski and Janine Peterson from Harpeth Hall School, Nashville, TN. Shaun Rancatore The man's statement was based on the fact that: The first point marked by the two people was the point at the right angle of a right triangle who's point also consist of the tree and the next point marked by the people 20 feet away from the first point along the chasm. The women stays at that second point which is at the intersection of the hypotenuse and the base of the triangle mentioned above. It is also the intersection of the hypotenuse and the base of a second right-triangle who's points are(starting at the right angle point) the third point where the man walked to(alone), which is 20 feet away from the point where the woman stayed, the point where the man walked to after turning 90 degrees, and the point where the woman is standing. The vertical angles at the point where the woman is standing, which are formed by the intersection of the line which is the two hypotenuse combined and the line which is the two bases combined are congruent because vertical angles are congruent. The bases of the two triangles are equal because the people measured both of them to be 20 ft. And the right angles of the two triangles are equal because 90degrees=90degrees. Knowing these three facts allows us to deduce the fact that the triangles are congruent, which in turn leads us to the fact that their sides are equal because the bases are equal. Since the sides are equal than the sides opposite of the corresponding hypotenuse in each triangle are equal. >From this point all the man had to do was measure the distance between him and the edge of the chasm and that would give him the length of the other side opposite of the hypotenuse and the base of the other triangle, which is the distance across the chasm. By the way, the reason the point at which the man stopped walking would be the intersection of the hypotenuse and the side that is not the base is because if the woman blocked his view of the tree that would mean she is on the same line as the tree and the man it also means he closed the triangle. -Shawn- Aron Freidenreich Problem of the Week: March 20-24 Aron Freidenreich Akiba Hebrew Academy The lines created from the "spot marked on the edge of the chasm directly across from the tree" and the 20 paces they walk along the edge of the chasm are perpendicular, forming a 90 degree angle *. Then, the boy walks an extra 20 paces, congruent to the first 20. From there, he makes a 90 degree turn, and then walks until he is a point on the line formed by the girl and the tree on the other side of the chasm. So, there are two right triangles involved: one with the three points: 1) the tree, 2) the point directly across from the tree, and 3) the girl; and the other triangle with the points: 1) the boy, 2) the point where he made a 90 degree turn, and 3) the girl. The 2 angles formed near the girl are congruent by the vertical angle theorem. Then, the two 20 pace distances are congruent. And finally, the 2 90 degree angles are congruent. So, the 2 triangles are congruent by ASA. Since corresponding sides of congruent triangles are congruent, the distance the boy walked away from the chasm, and the distance between the tree on the other side of the chasm and the point directly across from it are congruent. Therefore, a rope with the length of one distance will equal the length of the other one, allowing them to get across. * This is because the shortest distance across the chasm is a line perpendicular to its edge. Perpendicular lines form a 90 degree angle. Avi Klein Avi Klein Akiba Hebrew Academy 10th Grade We know that the two lower segments (opposite side of the chasm from the tree) are equal because he walks equal distances. We know that the angle leading to the tree equals 90 degrees because it is the shortest distance. Then using the vertical angle theorem, we can prove triangles congruent by ASA and then because corresponding sides of congruent triangles are congruent, you have your answer. Eric Wahl Annie, Here is a response from Eric Wahl Grade 11 Masterman Philadelphia, Pa Let S be the starting point, T, the tree, G, the girl's location 20 paces down the road, A, the guy's initial location 20 paces further down the road, and B, the final location of the guy. Right triangles TSG and BAG are congruent (LA- SG = AG and David Love The problem asks how some boy and girl figured how to cross a chasm. By retracing their steps we can create two triangles and show that two sides are also congruent, in order to prove the theory used by the two hero's. First you start off with a point, representing where the hero's start out. Then you make another point some distance from it, and connect the two lines. That line represents the distance across the chasm. From the original point we create a line of 20 paces that is 90 degrees from the other line. That line represents the distance that both the hero's walked, at the beginning, along the side of the chasm. From there you make a point and continue that line another 20 paces, representing the distance that the male hero covered on his own. From there you create another 90 degree angle that the male hero make, and create a line heading away from the chasm. To determine how long that line is, you must first create a line from the other side of the chasm (the tree), and continue it so that it passes through the point where the girl is standing. You then continue that line, and the other line until they intersect. When all that is done you should have to triangles. You can prove that they are congruent by ASA theorem. First of all both triangles have right angles (the 90 degrees the boy turned, and the 90 degrees from the tree to the distances the boy and girl both walked). Second they both have sides of 20 paces (the one the girl and the boy walked, and the one just the boy walked.) When the first line that the boy created, and the imaginary line that the boy sees at the end intersect each other, They create angles which are equal to each other, as proven by the vertical angle theorem. That is enough to prove the triangles congruent. Because they are congruent the distance from the tree to the originally marked spot, equals the distance the boy walked by himself at the end. Therefore all he had to do was measure that and he knew by the fact that congruent triangles yield congruent line segments, that the two lengths would be equal. The picture is not drawn to scale but it shows how the triangles are formed, and you're able to predict how they how congruent sides --------- |90 / | / David Love | / Akiba Hebrew Academy 20 | / 10th grade | / | / |n/ |/ / /| /n|20 / | / | / 90| ------ Ryan M. Howley Okay, the tree is point B, and they start at Point A. When they walk along the chasm, they walk in a line perpendicular to AB. She stops at point C, 20 paces from AB, but he continues in the same line to point D. Then, he turns 90 degrees and walks twenty paces to point E. Now, angle CAB is a right angle, along with EDC. Since alt-int angles are congruent, ED is // to AB. Then, using alt-int angles again, angle E is congruent to angle B. Then, triangle DCE is congruent to ACB because of angle-side-angle. Therefore, ED is congruent to BA because of CPCTC. So, the length of BA is 20 paces. Daniel A. Gabriel Daniel A. Gabriel Akiba Hebrew Academy, 10th Grade Lets start by labeling everything for easier reference. Label the tree Z. Label the point directly opposite from the tree X. Label the point where she stops walking, 20 paces alongside the chasm away from point X, C. When he continues walking for 20 more step and then stops he stops at point V. When he turns 90 degrees and walks away from the chasm, he stops at point B. Now we need to fill in some auxiliary line to see triangles which will help us find the answer. Draw Auxiliary line BZ and ZX. You know that angle ZXC is 90 degrees because it is directly across from the tree and the shortest distance across the chasm is the perpendicular distance. You now have 2 right triangles, ZXC and BVC. You know that VC and CX both equal 20 so they are congruent. Angle VCB= angle ZCX because all vertical angles are equal. You now have 2 angles and the included side of two triangles congruent so by ASA the two triangles are congruent. If that is the case then corresponding parts of the triangles are equal which means that VB=ZX and that answers your question. Marc Romanoff Problem of the Week March 20-24 Marc Romanoff Akiba Hebrew Academy 10th Grade Draw 2 parallel lines representing the chasm. If it wasn't parallel, the could cross it where it intersects. At one point, mark off a starting point, call it A. Opposite point A is the tree. Moving 20 units to the left of A, mark point X. 20 units to the left of X, mark point N. At N make a 90 degree angle, perpendicular to the chasm. At one point, he can see the tree. By drawing a line from this point to the tree you create two triangles which are congruent by ASA (vert angles =), (90 degree angles equal), (NX=XA). Therefore, by corresponding parts of congruent triangles are congruent, the distance walked from the edge of the chasm equals the distance of the chasm. This is all assuming, of course, that they don't slip into the chasm during the measuring. Christina Hughes and Teri Hreha Solution to POW for March 20-24 Submitted by Christina Hughes and Teri Hreha,Steel Valley High School, Pa. The guy and girl stand directly across from the tree. They both walk twenty paces along the chasm. The girl stays there. Then the guy walks another 20 paces. He them turns 90 degrees and walks away from the chasm until she he is directly in the line of sight between him and the tree. If you draw a diagram you can see two triangles; they look congruent, but you can't tell. The shorter sides of the triangles are both congruent because they're both 20. Both angles are 90 degrees; then there are vertical angles in between them; so that means that the triangles are congruent by ASA. So the longer sides of each triangle are the same length. Beth Skwarecki Solution to POW for March 20-24 Submitted by Beth Skwarecki, Steel Valley High School, Pa. The 20 paces, a 90 degree angle, and vertical angles are the same in both triangles. The two triangles are congruent by ASA, so "z" must be equal to "y." Ashley Hall, Kristin Biermann, and Bridget Coyle Ashley Hall & Kristin Biermann & Bridget Coyle Mt. St. Joseph Academy Grade 10 [they submitted their answer as a Sketchpad sketch. I have extracted the text.] Why is the segment between point A and point T congruent to the segment between point B and point C? We found that these two segments are congruent because of they are corresponding parts of congruent triangles. The congruent triangles are triangle ATG and BCG. We found that these are congruent triangles by Angle side angle. Angle CGB is congruent to angle TGA because vertical angles are congruent. Segment AG is congruent to segment BG because point G is the midpoint of segment AB. Angle CBG is congruent to angle TAG because they are both right angles. They are know to be right angles because both segment TA and segment CB are perpendicular to segment AB. Therefore, segment AT is congruent to segment BC. Jordan Walsh and Kate Devine Jordan Walsh and Kate Devine, Grade 9, Mt. St. Joseph's Academy Statements & Reasons 1.ang. TPG=90;ang. GCB=90 1. given 2.ang. TPG cong. ang GCB 2.substitution 3.ang. TGP & CGB are vert. ang. 3.def. of vert. ang. 4.ang. TGP & CGB are cong. 4.vert. ang. are cong. 5.PG is cong. PC 5.given 6.tri. TPG is cong. tri. BCG 6.A.S.A. post. 7.CB is cong. TB 7.C.P.C.T.C. Previous page || Next problem || Previous problem || Table of Contents || Forum Home Page