For a lezim kavayat the girls of standard IX are arranged in two circular concentric rings as shown in the figure if PA = 15 m, PR = 25 m, QR = 10 m, then the measures of RS, PQ, QT are

# For a lezim kavayat the girls of standard IX are arranged in two circular concentric rings as shown in the figure if PA = 15 m, PR = 25 m, QR = 10 m, then the measures of RS, PQ, QT are

1. A
20, $15\sqrt{5}$, 30
2. B
40, $5\sqrt{5}$, 60
3. C
40, $15\sqrt{5}$, 60
4. D
None of these

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### Solution:

Consider ∠PAR and ∠PAQ from the triangles PAQ and PAR.
Both are perpendicularly drawn from the center of the circle to the chord RS.
${\mathit{PR}}^{2}={\mathit{PA}}^{2}+{\mathit{AR}}^{2}$
$⇒{25}^{2}={15}^{2}+{\mathit{AR}}^{2}$
$⇒{\mathit{AR}}^{2}=625-225=400={20}^{2}$
$⇒\mathit{AR}=20$
RS = 2AR = 40
QA = QR + AR = 10 + 20 = 30
QT = 2QA = 60
In triangle PQA,
${\mathit{PQ}}^{2}={\mathit{PA}}^{2}+{\mathit{QA}}^{2}$
$⇒{\mathit{PQ}}^{2}={15}^{2}+{30}^{2}$
$⇒{\mathit{PQ}}^{2}=225+900=1125={\left(15\sqrt{5}\right)}^{2}$
$⇒\mathit{PQ}=15\sqrt{5}$
So, Option 3 is correct.

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