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In the figure, there are two concentric circles with centre O. $ P R T $ and $ P Q S $ are tangents to the inner circle from a point $ P $ lying on the outer circle. If $ P R=5 \mathrm{~cm} $, find the length of $ P S $.
"
Given:
In the figure, there are two concentric circles with centre O. \( P R T \) and \( P Q S \) are tangents to the inner circle from a point \( P \) lying on the outer circle.
\( P R=5 \mathrm{~cm} \).
To do:
We have to find the length of \( P S \).
Solution:
Join $OS$ and $OP$.
In $\triangle POS$,
$PO = OS$
This implies,
$\triangle POS$ is an isosceles triangle.
We know that,
In an isosceles triangle, if a line drawn perpendicular to the base of the triangle from the common vertex of the equal sides, then that line will bisect the base.
$PQ = PR = 5\ cm$ ($PRT$ and $PQS$ are tangents to the inner circle from a point $P$ lying on the outer circle)
$PQ = QS$
$PQ = 5\ cm$
$QS = 5\ cm$
From the figure,
$PS = PQ + QS$
$PS = 5 + 5$
$PS = 10\ cm$
The length of \( P S \) is $10\ cm$.
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