Question

Write a paragraph proof to finish proving the converse of the Pythagorean Theorem.

Answer 1

Answer:

A Proof for the Converse of the Pythagorean Theorem. Converse of the Pythagorean Theorem: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

Answer 2

Answer:

The converse of the Pythagorean Theorem is:

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

That is, in ΔABC, if c2=a2+b2 then ∠C is a right triangle, ΔPQR being the right angle.

We can prove this by contradiction.

Let us assume that c2=a2+b2 in ΔABC and the triangle is not a right triangle.

Now consider another triangle ΔPQR. We construct ΔPQR so that PR=a, QR=b and ∠R is a right angle.

By the Pythagorean Theorem, (PQ)2=a2+b2.

But we know that a2+b2=c2 and a2+b2=c2 and c=AB.

So, (PQ)2=a2+b2=(AB)2.

That is, (PQ)2=(AB)2.

Since PQ and AB are lengths of sides, we can take positive square roots.

PQ=AB

That is, all the three sides of ΔPQR are congruent to the three sides of ΔABC. So, the two triangles are congruent by the Side-Side-Side Congruence Property.

Since ΔABC is congruent to ΔPQR and ΔPQR is a right triangle, ΔABC must also be a right triangle.

This is a contradiction. Therefore, our assumption must be wrong.

hope it helps you dear friend

#Nikki