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Seg PM is a median of ∆PQR. If PQ = 40, PR = 42 and PM = 29, find QR.

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#### Solution

In ∆PQR, point M is the midpoint of side QR.

\[{PQ}^2 + {PR}^2 = 2 {PM}^2 + 2 {QM}^2 \left( \text{by Apollonius theorem} \right)\]

\[ \Rightarrow \left( 40 \right)^2 + \left( 42 \right)^2 = 2 \left( 29 \right)^2 + 2 {QM}^2 \]

\[ \Rightarrow 1600 + 1764 = 1682 + 2 {QM}^2 \]

\[ \Rightarrow 3364 - 1682 = 2 {QM}^2 \]

\[ \Rightarrow 1682 = 2 {QM}^2 \]

\[ \Rightarrow {QM}^2 = 841\]

\[ \Rightarrow QM = 29\]

\[ \Rightarrow QR = 2 \times 29\]

\[ \Rightarrow QR = 58\]

Hence, QR = 58.

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