Question

In triangle $ ABC $, the length of sides are $ AB = 7 $, $ BC = 10 $, and $ AC = 5 $. What is the length of the median drawn from vertex $ B $?

• A. 6
• B. 5
• C. 7
• D. 8

Hint:

The median from vertex \( A \) to side \( BC \) in triangle is: \[ m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2} \] Use correct labeling of triangle sides to avoid confusion.

Answer 1

The Correct Option is A

To find the length of the median from vertex \( B \) to side \( AC \), we use Apollonius's theorem: \[ m_b = \frac{1}{2} \sqrt{2AB^2 + 2BC^2 - AC^2} \] Substituting the given values: \[ m_b = \frac{1}{2} \sqrt{2(7)^2 + 2(10)^2 - (5)^2} \] \[ m_b = \frac{1}{2} \sqrt{2 \times 49 + 2 \times 100 - 25} \] \[ m_b = \frac{1}{2} \sqrt{98 + 200 - 25} \] \[ m_b = \frac{1}{2} \sqrt{273} \] Calculating the numerical value: \[ \sqrt{273} \approx 16.52 \quad \Rightarrow \quad m_b \approx \frac{16.52}{2} = 8.26 \] The closest integer value is \boxed{8}, which corresponds to option (D).