Prove Converse of Pythagorean Theorem

Joined
Jun 27, 2021
Messages
5,386
Reaction score
422
College Algebra
Chapter 1/Section 3

Prove the converse of the Pythagorean Theorem. Explain each step along the way.
 
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 c^2=a^2+b^2 then ∠C is a right triangle, ΔPQR being the right angle.

proof;
We can prove this by contradiction.

Let us assume that c^2=a^2+b^2 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.

f-conv_pyth_1_2_1.gif


By the Pythagorean Theorem, (PQ)^2=a^2+b^2.

But we know that a^2+b^2=c^2 and c=AB.

So, (PQ)^2=a^2+b^2=(AB)^2.

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

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

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.
 
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 c^2=a^2+b^2 then ∠C is a right triangle, ΔPQR being the right angle.

proof;
We can prove this by contradiction.

Let us assume that c^2=a^2+b^2 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.

f-conv_pyth_1_2_1.gif


By the Pythagorean Theorem, (PQ)^2=a^2+b^2.

But we know that a^2+b^2=c^2 and c=AB.

So, (PQ)^2=a^2+b^2=(AB)^2.

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

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

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.

Great study notes. A job well-done!
 

Members online

No members online now.

Forum statistics

Threads
2,521
Messages
9,844
Members
697
Latest member
RicoCullen
Back
Top