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. 4 : 3

Hint:

Use the median length formula for each side and add all three before forming the ratio.

Answer 1

The Correct Option is D

Step 1: Recall the median length formula.
For a triangle with sides \( a, b, c \), the median to side \( a \) has length \( m_a \), where \( m_a^2 = \dfrac{2b^2 + 2c^2 - a^2}{4} \).
Similar formulas hold for \( m_b \) and \( m_c \).

Step 2: Add the three median formulas.
\( m_a^2 + m_b^2 + m_c^2 = \dfrac{(2b^2+2c^2-a^2) + (2a^2+2c^2-b^2) + (2a^2+2b^2-c^2)}{4} \)
The numerator simplifies to \( 3a^2 + 3b^2 + 3c^2 \).

Step 3: Simplify the sum of squares of medians.
\( m_a^2 + m_b^2 + m_c^2 = \dfrac{3(a^2+b^2+c^2)}{4} \).

Step 4: Form the required ratio.
Sum of squares of sides \( = a^2+b^2+c^2 \).
Ratio \( = (a^2+b^2+c^2) : \dfrac{3(a^2+b^2+c^2)}{4} = 1 : \dfrac{3}{4} = 4 : 3 \).

Final Answer:
The ratio of sum of squares of sides to sum of squares of medians is 4 : 3. \[ \boxed{4 : 3} \]