Chapter 10, Problem 9SGR

To determine

To state whether the given statement is true or false.

Answer to Problem 9SGR

True

Explanation of Solution

Given:

The converse of the Pythagorean Theorem is true.

Concept used:

Converse of Pythagorean Theorem states that:

If the length of a triangle is $a,b$ and $c$ and $c^2=a^2+b^2$ , then the triangle is a right - angle triangle.

Calculation:

Let $c^2=a^2+b^2$ in $\Delta ABC$ and the triangle is not a right triangle.

Now consider another triangle $\Delta PQR$ .

  $\Delta PQR$ is such that $PR=a,QR=b$ and $\angle R$ is a right angle.

By the Pythagorean Theorem,

  ${\left(PQ\right)}^2=a^2+b^2$

But $a^2+b^2=c^2$ and $c=AB$ .

So, ${\left(PQ\right)}^2=a^2+b^2={\left(AB\right)}^2$

  $\Rightarrow {\left(PQ\right)}^2={\left(AB\right)}^2$

Since $PQ$ and $AB$ are length of sides, taking square root on both sides,

  $\Rightarrow PQ=AB$

That is, all the three sides of $\Delta PQR$ are congruent to the three sides of $\Delta ABC$ .

So, the two triangles are congruent by the Side-Side-Side Congruence Property.

Since $\Delta ABC$ is congruent to $\Delta PQR$ and $\Delta PQR$ is a right triangle,

Therefore, $\Delta ABC$ must also be a right triangle.

This is a contradiction.

Therefore, assumption was wrong.

Conclusion:

Therefore, the given statement is true.