Question

(08.01 HC)
Prove the Converse of the Pythagorean Theorem using similar triangles. The Converse of the Pythagorean Theorem states that when the sum of the squares of the lengths of the legs of the triangle equals the squared length of the hypotenuse, the triangle is a right triangle. Be sure to create and name the appropriate geometric figures. (10 points)

Answer 1

Hope this helps you

Answer 2

Lets say we have a triangle ABC and it's right triangle because it has 90 degrees angle. The longest side of the angle or the opposite side of a right angle AB is a hypotenuse.

Lets call the angle AC as segment a

The Angle BC as segment b

The angle AB (hypotenuse) as segment c

Now I want to show the relationship  between them by constructing another segment between C and the hypotenuse, I want them to intersect at the right angle and call another new point as a D.

The reason why I did that because now I can show relationship between similar triangles.

Now we have three Triangle

Triangle ABC

Triangle DBC

Triangle ADC

ADC is the similar to ABC because both of them have a right angle and they both share angle DAC

△ADC~△ACB

AC/AB = AD/AC        (and if we rewrite it properly)

a/c = d/a                    (cross multiply)

a² = cd

△BDC~△BCA

BC/BA = BD/BC        (and if we rewrite it properly)

b/c=e/b                      (cross multiply)

b² = ce

Now a² + b² = cd + ce

cd + ce = c (d+e)

* If d and e is a part of a segment c our equation is actually

c(c) = c²

Now our relationship is  a² +b² =c² we just proved the Pythagorean Theorem.