Question:
Segment \( PM \) is a median of \( \triangle PQR \), \( PM = 9 \), and \( PQ^2 + PR^2 = 290 \). Find \( QR \).
Segment \( PM \) is a median of \( \triangle PQR \), \( PM = 9 \), and \( PQ^2 + PR^2 = 290 \). Find \( QR \).
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The Apollonius theorem is useful for finding side lengths in triangles when medians are involved.
Updated On: Nov 12, 2025
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Solution and Explanation
Step 1: Using the Apollonius theorem for medians in a triangle:
\[
PQ^2 + PR^2 = 2PM^2 + \frac{1}{2}QR^2.
\]
Step 2: Substituting the given values \( PQ^2 + PR^2 = 290 \), \( PM = 9 \):
\[
290 = 2(9^2) + \frac{1}{2}QR^2.
\]
Step 3: Simplify:
\[
290 = 2(81) + \frac{1}{2}QR^2.
\]
\[
290 = 162 + \frac{1}{2}QR^2.
\]
Step 4: Rearrange and solve for \( QR^2 \):
\[
290 - 162 = \frac{1}{2}QR^2.
\]
\[
128 = \frac{1}{2}QR^2 \quad \Rightarrow \quad QR^2 = 256.
\]
Step 5: Taking the square root:
\[
QR = \sqrt{256} = 16.
\]
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