Need help with a geometry question...

Question

The question is...

Andre and Clare are discussing triangle ABC that has a right angle at C and a hypotenuse of length 15 units. Andre thinks the triangle could have legs that are 9 and 12 units long. Clare thinks angle B could be 20 degrees and then side BC would be 14.1 units long. Do you agree with either of them? Explain or show your reasoning.

Thank you for those who help

Answer 1

If the sides of a right triangle are 3,4,5 this is known as a Pythagorean Triple, because 3² + 4² = 5².

Since the sides in the given problem (9, 12, 15) are multiples of the known 3,4,5, then 9, 12, 15 is also a Pythagorean Triple. So I agree with Andre.

If you have a right triangle with a hypotenuse of length 15 (and that is all you know), then angle B could be 20 degrees and angle A would then have to be 70 degrees. In that case using right triangle trigonometry, side BC would equal 15 cos(20) ≅ 14.1. Using similar logic side AC would have a length about 5.13. The Pythagorean Theorem would still apply (14.1² + 5.13² = 15²). So Clare is correct too. As a matter of fact there are infinitely many right triangles with a hypotenuse of 15.

This Desmos graph shows a couple other possibilities:

desmos.com/calculator/bt3esw5hp9