If S is a point on side PQ of a △PQR such that PS = QS = RS, then
a. PR . QR = RS²
b. QS² + RS² = QR²
c. PR² + QR² = PQ²
d. PS² + RS² = PR²

Solution:

Given, S is a point on side PQ of a triangle PQR.

Also, PS = QS = RS

In triangle PSR,

Given, PS = RS

We know that angles opposite to equal sides in a triangle are equal

∠P = ∠R

So, ∠P = ∠1

Similarly, in triangle RSQ

Given, RS = QS

∠R = ∠Q

So, ∠Q = 2

Considering triangle PQR,

We know that the sum of all three interior angles of a triangle is always equal to 180 degrees.

So, ∠P + ∠Q + ∠PRQ = 180°

∠1 + ∠2 + ∠PRQ = 180°

AAA criterion states that if two angles of a triangle are respectively equal to two angles of another triangle, then by the angle sum property of a triangle their third angle will also be equal.

So, ∠PRQ = ∠1 + ∠2

Now, ∠1 + ∠2 + ∠1 + ∠2 = 180°

2(∠1 + ∠2) = 180°

∠1 + ∠2 = 180°/2

∠1 + ∠2 = 90°

∠PRQ = 90°

So PRQ is a right triangle with right angle at R.

By Pythagoras theorem,

PQ² = PR² + QR²

Therefore, option C is true.

✦ Try This: PQR is a right triangle, right angled at Q and QS ⊥ PR. If PQ = 6 cm and PS = 4 cm, then find QS, RS and QR

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

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NCERT Exemplar Class 10 Maths Exercise 6.1 Problem 12

If S is a point on side PQ of a △PQR such that PS = QS = RS, then, a. PR . QR = RS², b. QS² + RS² = QR², c. PR² + QR² = PQ², d. PS² + RS² = PR²

Summary:

If S is a point on side PQ of a △PQR such that PS = QS = RS, then PR² + QR² = PQ²

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