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.
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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,Quote:
Originally Posted by MATNTRNG
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?
Yes,Quote:
Originally Posted by MATNTRNG
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
