Question

4.5 the converse of the pathagorean theorem​

Answer 1

If a^2 + b^2 = c^2, then the triangle is a right triangle.

Answer 2

The converse of the Pythagorean Theorem,
`{\bf (PQ)}^(\bf 2)={\bf a^2+b^2}` is true

Explanation:

The Pythagorean theorem or Pythagoras's theorem is a statement about the sides of a right triangle.

If the lengths of the legs are a and b, and the length of the hypotenuse is c, then,
`a^(2)+b^(2)=c^(2)`

To prove that the converse of the Pythagorean Theorem,
`(PQ)^2=a^2+b^2`

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.

PQ=AB

That is, all the three sides of
`\triangle PQR` are congruent to the three sides of
`\triangle ABC` . So, the two triangles are congruent by the Side-Side-Side Congruence Property.

Since
`\triangle ABC` is congruent to
`\triangle PQR` and
`\triangle PQR` is a right triangle,
`\triangle ABC` must also be a right triangle.

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

Therefore the converse of the Pythagorean Theorem,

`(PQ)^2=a^2+b^2`