In $△ P Q R$ , point $S$ is the midpoint of side $Q R$ . If $P Q = 11, P R = 17, P S = 13$ , find $Q R$ .

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Solution

In $△ P Q R$ , point $S$ is the midpoint of side $Q R$ . [Given]
$∴$ seg $P S$ is the median.
$∴ {P Q}^{2} + {P R}^{2} = 2 {P S}^{2} + 2 {S R}^{2}$ [Apollonius theorem]
$∴ 11^{2} + 17^{2} = 2 (13)^{2} + 2 {S R}^{2}$
$∴ 121 + 289 = 2 (169) + 2 S R^{2}$
$∴ 410 = 338 + 2 {S R}^{2}$
$∴ 2 {S R}^{2} = 410 - 338$
$∴ 2 {S R}^{2} = 72$
$∴ {S R}^{2} = \frac{72}{2} = 36$
$∴ S R = \sqrt{36}$ [Taking square root of both sides]
$= 6$ units Now, $Q R = 2 S R$ [S is the midpoint of $Q R$ ] $= 2 \times 6$
$∴ Q R = 12$ units

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△ P Q R

S

Q R

P Q = 11

P R = 17, P S = 13

Q R

In $△ P Q R$ , point $S$ is the midpoint of side $Q R$ . If $P Q = 17, P R = 11, P S = 13.$ Find $Q R$

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