Question

In a triangle ABC, if \( BC = 5 \), \( CA = 6 \), \( AB = 7 \), then the length of the median drawn from \( B \) onto \( AC \) is:

• A. \( 5 \)
• B. \( \sqrt{7} \)
• C. \( \sqrt{5} \)
• D. \( 2\sqrt{7} \)

Hint:

The length of the median in a triangle can be found using Apollonius's theorem, which simplifies the process of calculating the median when the sides of the triangle are known.

Answer 1

The Correct Option is D

We are given a triangle \( ABC \) with sides \( BC = 5 \), \( CA = 6 \), and \( AB = 7 \). We are asked to find the length of the median drawn from \( B \) onto \( AC \). 
Step 1: The formula for the length of the median from vertex \( B \) to side \( AC \) in any triangle is given by the Apollonius's theorem: \[ m_{BC}^2 = \frac{2AB^2 + 2AC^2 - BC^2}{4} \] where \( m_{BC} \) is the length of the median from \( B \) onto \( AC \), and \( AB \), \( AC \), and \( BC \) are the sides of the triangle. 
Step 2: Substitute the values of \( AB = 7 \), \( AC = 6 \), and \( BC = 5 \) into the formula: \[ m_{BC}^2 = \frac{2(7^2) + 2(6^2) - 5^2}{4} \] \[ m_{BC}^2 = \frac{2(49) + 2(36) - 25}{4} \] \[ m_{BC}^2 = \frac{98 + 72 - 25}{4} \] \[ m_{BC}^2 = \frac{145}{4} \] \[ m_{BC} = \frac{\sqrt{145}}{2} = 2\sqrt{7} \] Thus, the length of the median from \( B \) to \( AC \) is \( 2\sqrt{7} \).