Question

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?

• A. 1:2
• B. 2:1
• C. 2:3
• D. 3:4
• E. 4:3

Answer 1

Answer: (E)

The correct answer is option (E): 4:3

Let's analyze this geometry problem. We want the ratio between the sum of the squares of the sides of a triangle and the sum of the squares of its medians.

Let the sides be a, b, c and the medians be m_a, m_b, m_c. We need the two quantities:

1. Sum of squares of the sides: a2 + b2 + c2
2. Sum of squares of the medians: ma2 + mb2 + mc2

Apply Apollonius's theorem to each median:

• For ma: b2 + c2 = 2(ma2 + (a/2)2) →  4ma2 = 2b2 + 2c2 - a2
• For mb: a2 + c2 = 2(mb2 + (b/2)2)  →  4mb2 = 2a2 + 2c2 - b2
• For mc: a2 + b2 = 2(mc2 + (c/2)2)  →  4mc2 = 2a2 + 2b2 - c2

Sum the three equations:

4(ma2 + mb2 + mc2) = 3(a2 + b2 + c2)

Therefore
ma2 + mb2 + mc2 = (3/4)(a2 + b2 + c2)

The required ratio is
(a2 + b2 + c2) / (ma2 + mb2 + mc2) = 1 / (3/4) = 4/3

Hence the ratio is 4:3.