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Prove that if in a triangle square on one side is equal to the sum of the squares on the other two sides, then the angle opposite the first side is a right angle

Solution:

Given that , in a triangle the square of one side is equal to the sum of the squares on the other two sides.

We have to prove that the angle opposite to the first side is a right angle.

Let the given triangle be ABC.

As per given statement,

AC² = AB² + BC² ------------------------ (1)

Let us consider another triangle PQR with right angle at Q.

Now, PQ = AB

RQ = BC

By using pythagoras theorem in △PQR,

PR² = PQ² + RQ²

Substituting the value of PQ and RQ,

PR² = AB² + BC² -------------------- (2)

Comparing (1) and (2),

PR² = AC²

Taking square root on both sides,

PR = AC ----------------------------------- (3)

Now considering △ABC and △PQR,

AB = PQ

BC = RQ

From (3), AC = PR

SSS criterion states that if three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.

By SSS criterion, △ABC ⩬ △PQR

So, ∠B = ∠Q

We know, ∠B = 90°

So, ∠Q = 90°

Therefore, it is proven that the angle opposite to the first side is a right angle.

✦ Try This: Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 6

NCERT Exemplar Class 10 Maths Exercise 6.4 Sample Problem 2

Prove that if in a triangle square on one side is equal to the sum of the squares on the other two sides, then the angle opposite the first side is a right angle

Summary:

It is proven that if in a triangle, square on one side is equal to the sum of the squares on the other two sides then the angle opposite the first side is a right angle

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