Thread: 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.

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,







Therefore



from which the result follows

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?

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



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.

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