Question 32

Question 32 Students in a class using their knowledge of the Pythagorean theorem to make conjectures about triangles. A student makes the conjecture shown below: A triangle has side lengths x, y, and z. If x < y < z and x2 + y2 < z2, the triangle is an obtuse triangle. Use the Pythagorean theorem to develop a chain of reasoning to justify or refute the conjecture. You must demonstrate that the conjecture is always true or that there is at least one example in which the conjecture is not true. Students might not have a calculator for this one. A grid is also not provided so if the students wanted to draw a picture, they would need to use their scratch paper. They will have to use words to describe their triangle to make it fit.

Question 32 Picture a triangle with the side of length x on the bottom, the side of length y on the left, and the side of length z on the top. If x2 + y2 = z2, the triangle is a right triangle. Since x2 + y2 < z2, the sides with length x and y are left so they make a right triangle and the side with length z extends past the side with length y, if it starts at side with length x. The direction of side y would have to change so that it met the end of the line of side z. When this happens, the triangle will become an obtuse triangle, so the conjecture is true. This is a modified explanation from the answer guide. Pretty much, it just states that it is true because one side has to be much larger than the other side. In a right triangle, the sides would equal each other. Since the last side, the longest side, is longer than the other two sides, the triangle has to shift, creating an obtuse triangle. On the next slide, I”ll show a picture.

Question 32 z z x x y y The first triangle is a right triangle. If you square the two sides, it will equal the square of the third side. For the next triangle, I used the same sizes of x and y but extended z. You can see that since z is longer than y, the triangle will have to readjust in order to make it fit. Since when you do that, it has to extend past the 90 degree mark, the triangle becomes obtuse (larger than 90 degrees)