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MCQ

Fill in the Blanks

If S is a point on side PQ of a ΔQR such that PS = QS = RS, then ______.

#### Options

PR . QR = RS

^{2}QS

^{2}+ RS^{2}= QR^{2}PR

^{2}+ QR^{2}= PQ^{2}PS

^{2}+ RS^{2}= PR^{2}

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#### Solution

If S is a point on side PQ of a ΔQR such that PS = QS = RS, then **PR ^{2} + QR^{2} = PQ^{2}**.

**Explanation:**

Given, in ∆PQR,

PS = QS = RS .......(i)

In ∆PSR, PS = RS ......[From equation (i)]

⇒ ∠1 = ∠2 .......(ii)

[Angles opposite to equal sides are equal]

Similarly, in ∆RSQ, RS = SQ

⇒ ∠3 = ∠4 .......(iii)

[angles opposite to equal sides are equal]

Now, in ∆PQR, sum of angles = 180°

⇒ ∠P + ∠Q + ∠P = 180°

⇒ ∠2 + ∠4 + ∠1 + ∠3 = 180°

⇒ ∠1 + ∠3 + ∠1 + ∠3 = 180°

⇒ 2(∠1 + ∠3) = 180°

⇒ ∠l + ∠3 = 180∘2 = 90°

∴ ∠R = 90°

In ∆PQR, by Pythagoras theorem,

PR^{2} + QR^{2} = PQ^{2}

Concept: Application of Pythagoras Theorem in Acute Angle and Obtuse Angle

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