Question

The vertices of a triangle are $A(3,4)$, $B(7,2)$ and $C(-2,-5)$. Find the length of the median through the vertex A.

• A. $\frac{\sqrt{111}}{2}$units
• B. $\frac{\sqrt{147}}{2}$units
• C. $\frac{\sqrt{137}}{2}$units
• D. $\frac{\sqrt{122}}{2}$units

Answer 1

The correct option is D $\frac{\sqrt{122}}{2}$units

A median of a triangle is a line segment that joins the
vertex of a triangle to the midpoint of the opposite side.
Mid point of two points ${(x}_1,y_1)$ and ${(x}_2,y_2)$ is calculated by the formula $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$
Since AD is the median, this means, D is the mid point of BC.
Using this formula, mid point of BC $=(\frac{7-2}{2},\frac{2-5}{2})=(\frac{5}{2},\frac{-3}{2})$

Distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ can be calculated using the formula $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Hence, length of AD $=\sqrt{{(\frac{5}{2}-3)}^2+{(\frac{-3}{2}-4)}^2}=\sqrt{\frac{1}{4}+\frac{121}{4}}=\sqrt{\frac{122}{4}}=\frac{\sqrt{122}}{2}$