Question : In the given figure, PQRS is a square inscribed in a circle of radius 4 cm. PQ is produced till point Y. From Y, a tangent is drawn to the circle at point R. What is the length (in cm) of SY?(Given that QR = QY)
Option 1: $4\sqrt{10}$
Option 2: $2\sqrt{10}$
Option 3: $6\sqrt{10}$
Option 4: $3\sqrt{5}$
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Correct Answer: $4\sqrt{10}$
Solution :
RQ = QY
The radius of the circle = 4 cm
Diameter = 2 × 4 = 8 cm
Let $a$ be the side of the square.
⇒ $a^2$ + $a^2$ = $8^2$
⇒ $a$ = $\sqrt{\frac{64}{2}}$ = $4\sqrt{2}$
In $\triangle$RYQ,
⇒ RQ = QY = $4\sqrt{2}$
In $\triangle$SPY,
⇒ SP = $4\sqrt{2}$
⇒ PY = $4\sqrt{2}$ + $4\sqrt{2}$ = $8\sqrt{2}$
By Pythagoras theorem,
⇒ SY
2
= SP
2
+ PY
2
⇒ SY
2
= $(4\sqrt2)^2$ + $(8\sqrt{2})^2$
⇒ SY
2
= 32 + 128 = 160
⇒ SY = $4\sqrt{10}$ cm
Hence, the correct answer is $4\sqrt{10}$.
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