Question

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

Answer 1

Answer:

Given:- ABC is a triangle

AC^2

=AB^2 +BC^2

To prove:- ∠B=90°

Construction:- Construct a triangle PQR right angled at Q such that, PQ=AB and QR=BC

Proof:-

In △PQR

PR ^2 =PQ ^2+QR ^2

(By pythagoras theorem)

⇒PR ^2 =AB ^2 +BC ^2 .....(1)

(∵AB=PQ and QR=BC)

AC ^2 =AB ^2 +BC ^2 .....(2)(Given)

From equation (1)&(2), we have

AC ^2

=PR ^2

⇒AC=PR.....(3)

Now, in △ABC and △PQR

AB=PQ

BC=QR

AC=PR(From (3))

∴△ABC≅△PQR(By SSS congruency)

Therefore, by C.P.C.T.,

∠B=∠Q

∵∠Q=90°

∴∠B=90°

Hence proved.

solution