Date: 2/8/96 at 9:47:26 From: Mrs. Doris E. Risenhoover Subject: Isosceles triangle I need to find the answer to the following problem: 1. an isosceles triangle whose points are A, B, & C. 2. One long side of the triangle has point p 3. The other side of the trangle has point q 4. The points p and q are located so that AC=AP=PQ=QB 5. I need to find angle B, and how to do problem
Date: 3/4/96 at 21:37:9 From: Doctor Elise Subject: Re: Isosceles triangle Hi! Okay, you have an isosceles triangle ABC. And somewhere on the sides, you have points Q and P. I have to assume here that AP and AQ both already lie on the sides of the triangle-- that is, you don't have point p on the line BC so that you have to draw AP across the middle of the triangle. So we draw an isosceles triangle ABC, and then we put P between A and B, and Q between A and C, and then we connect P and Q. Now we look at requirement 4, which says that AC = AP = PQ. This tells us that the triangle formed by APQ is equilateral, right? All the angles have to be 60 degrees in APQ, so we know that angle A is 60 degrees. Now we have isosceles triangle ABC, with angle A = 60. If A turns out to be one of the two equal angles in the isosceles triangle, then either C or B has to be the same, leaving the third angle to be 60 degrees also to make up 180 total. So all three angles would be 60 degrees. If angle A is not one of the two equal angles, then we subtract angle A from 180 degrees and divide the result by 2 to get B and C, and all three angles STILL end up being 60 degrees. ABC is an equilateral triangle! I hope this helps. P.S. I can draw an isosceles triangle, where P is between C and D, that meets all 5 of the requirements, but I can't solve for angle B. The solution above doesn't use the clue AP = QB, which makes me a little suspicious of my assumption. Any ideas? -Doctor Elise, The Math Forum
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