1. ## Right Triangle Proof

Triangle ABC is a right triangle with height h to the hypotenuse. The height divides the hypotenuse into segments a1 and b1. Prove that h^2 = a1 * b1 without using triangle similarity.

2. Originally Posted by MATNTRNG
Triangle ABC is a right triangle with height h to the hypotenuse. The height divides the hypotenuse into segments a1 and b1. Prove that h^2 = a1 * b1 without using triangle similarity.
Using Pythagoras' theorem,

$\displaystyle \left(a_1\right)^2+h^2=a^2$

$\displaystyle a^2+b^2=\left(a_1+b_1\right)^2$

$\displaystyle \left(b_1\right)^2+h^2=b^2$

Therefore

$\displaystyle \left(a_1\right)^2+h^2+\left(b_1\right)^2+h^2=\lef t(a_1+b_1\right)^2$

from which the result follows

3. If it is not known that triangle ABC is a right triangle but it is known that h^2 = a1 * b1, can it be proved that angle C must be a right angle? I know that the answer is yes. Any ideas as to how I would go about proving this?

4. Originally Posted by MATNTRNG
If it is not known that triangle ABC is a right triangle but it is known that h^2 = a1 * b1, can it be proved that angle C must be a right angle? I know that the answer is yes. Any ideas as to how I would go about proving this?
Yes,

as the non-right-angles are acute, we have

$\displaystyle a_1b_1=h^2\Rightarrow\frac{a_1b_1}{h}=h\Rightarrow \frac{b_1}{h}=\frac{h}{a_1}$

These ratios are the tangents of two angles, which being equal causes the angles to be equal.
Both inner triangles have a right-angle, therefore the third angles are also equal
and the two acute angles sum to 90 degrees.

5. ## right triangle proof

Hi guys,

Solution 0f followup question using Pythagorean Theorem and same side ids given h^2 =a1 x b1

h^2 + a1^2 =a^2
h^2 + b1^2 =b^2
2h^2 + a1^2 + b1^2 =a^2 + b^2
2a1xb1 + a1^2 + b1^2 = a^2 +b^2
(a1 + b1 )^2 = a^2 + b^2
triangle is a right triangle