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In the figure, \( O Q: P Q=3: 4 \) and perimeter of \( \Delta P O Q=60 \mathrm{~cm} \). Determine \( P Q, Q R \) and \( O P \).
"\n
Given:
In the figure, \( O Q: P Q=3: 4 \) and perimeter of \( \Delta P O Q=60 \mathrm{~cm} \).
To do:
We have to determine \( P Q, Q R \) and \( O P \).
Solution:
In the figure,
$OQ : PQ = 3:4$
Perimeter of $\triangle POQ = 60\ cm$
Let $OQ = 3x$ and $PQ = 4x$
In right angled triangle $OPQ$,
$OP^2 = OQ^2 + PQ^2$
$= (3x)^2 + (4x)^2$
$= 9x^2 + 16x^2$
$= 25x^2$
$= (5x)^2$
$\Rightarrow OP = 5x$
$OQ + QP + OP = 60\ cm$
$3x + 4x + 5x = 60$
$12x = 60$
$x = 5$
$PQ = 4x = 4 \times 5 = 20\ cm$
$QR = 2 OQ = 2(3x) = 6 \times 5 = 30\ cm$
$OP = 5x = 5 \times 5 = 25\ cm$
Therefore, $PQ=20\ cm, QR= 30\ cm$ and $OP=25\ cm$.