Question:easy

What is the ratio of the sum of the squares of the sides of a triangle to the sum of the squares of its median?

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Use the median length formula for each side and add all three before forming the ratio.
Updated On: Jul 21, 2026
  • 1 : 2
  • 2 : 1
  • 2 : 3
  • 4 : 3
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The Correct Option is D

Solution and Explanation

Step 1: Use a concrete right triangle to test the relationship.
Take a right triangle with sides $ a = 3 $, $ b = 4 $, $ c = 5 $ placed with the right angle at the vertex between $ a $ and $ b $.

Step 2: Compute the sum of squares of the sides.
$ 3^2 + 4^2 + 5^2 = 9 + 16 + 25 = 50 $.

Step 3: Compute each median using coordinates.
Place the right angle vertex at the origin, the other two vertices at $ (3,0) $ and $ (0,4) $, so the hypotenuse runs between them.
Midpoint of the hypotenuse is $ (1.5, 2) $, and the median from the origin has length $ \sqrt{1.5^2 + 2^2} = \sqrt{2.25+4} = \sqrt{6.25} = 2.5 $.
Midpoint of the side from $ (0,0) $ to $ (3,0) $ is $ (1.5,0) $, so the median from $ (0,4) $ has length $ \sqrt{1.5^2+4^2} = \sqrt{2.25+16} = \sqrt{18.25} $.
Midpoint of the side from $ (0,0) $ to $ (0,4) $ is $ (0,2) $, so the median from $ (3,0) $ has length $ \sqrt{3^2+2^2} = \sqrt{9+4} = \sqrt{13} $.

Step 4: Add the squares of the medians.
$ 2.5^2 + 18.25 + 13 = 6.25 + 18.25 + 13 = 37.5 $.

Step 5: Form the ratio and simplify.
$ 50 : 37.5 = 500 : 375 = 4 : 3 $, matching the general formula result.

Final Answer:
The ratio is 4 : 3. $ 4 : 3 $
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