Question

Write converse of pithagoras theorem and prove it​

Answer 1

$\fbox\pink{A}\fbox\blue{n}\fbox\purple{s}\fbox\green{w}\fbox\red{e}\fbox\orange{r}$

Converse of Pythagoras Theorem.

Statement:

In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Proof:

Here, we are given a triangle ABC in which Ac2=AB2+BC2.

We need to prove that ∠B=90o.

To start with, we construct a ΔPQR right-angled at Q such that PQ=AB and QR=BC.

Now, from Δ PQR, we have:

Pr2=PQ2+QR2           (Pythagoras Theorem, as ∠Q=90o)

or PR2=AB2+BC2        (By construction)........ (1)

But AC2=AB2+BC2         (Given).......... (2)

So, AC=PR           (From (1) and (2)).............. (3)

Now, in ΔABC and ΔPQR,

AB=PQ       (By construction)

BC=QR        (By construction)

AC=PR        (Proved in (3))

So, ΔABC≃ΔPQR         (SSS congruence)

∠B=∠Q                 (Corresponding angles of congruent triangles)

∠Q=90o                      (By construction)

So ∠B=90o