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

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Detailed Solution

Drawing a triangle with the sides

ABC

and the median

AD

through the vertex

A

will help us first solve this problem. The coordinates of point

D

will then be determined using the midpoint formula, and the length of the median

AD

will be determined using the distance formula.
According to the midpoint formula,

x = \frac{x_{1} + x_{2}}{2}

, and

y = \frac{y_{1} + y_{2}}{2}

The distance is calculated using the formula

\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}

Using this formula, midpoint of BC =(

\frac{7 - 2}{2}, \frac{2 - 5}{2}) = (\frac{5}{2}, \frac{- 3}{2})

Hence the 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}

units
Hence option 4 is correct.

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