Answer:
The converse of the Pythagorean Theorem, [tex]{\bf (PQ)}^{\bf 2}={\bf a^2+b^2}[/tex] is true
Step-by-step 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, [tex]a^{2}+b^{2}=c^{2}[/tex]
To prove that the converse of the Pythagorean Theorem, [tex](PQ)^2=a^2+b^2[/tex]
By the Pythagorean Theorem, [tex](PQ)^2=a^2+b^2[/tex]
But we know that [tex]a^2+b^2=c^2[/tex] and [tex]c=AB[/tex]
So, [tex](PQ)^2=a^2+b^2=(AB)^2[/tex]
That is, [tex](PQ)^2=(AB)^2[/tex]
Since PQ and AB are lengths of sides, we can take positive square roots.
PQ=AB
That is, all the three sides of [tex]\triangle PQR[/tex] are congruent to the three sides of [tex]\triangle ABC[/tex] . So, the two triangles are congruent by the Side-Side-Side Congruence Property.
Since [tex]\triangle ABC[/tex] is congruent to [tex]\triangle PQR[/tex] and [tex]\triangle PQR[/tex] is a right triangle, [tex]\triangle ABC[/tex] must also be a right triangle.
This is a contradiction. Therefore, our assumption must be wrong.
Therefore the converse of the Pythagorean Theorem,
[tex](PQ)^2=a^2+b^2[/tex]
