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Find the length of the median through $( - 2, - 5)$ of the triangle whose vertices are $( - 6,2)$,$(2, - 2)$, and $( - 2, - 5)$

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Last updated date: 18th Jun 2024
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Answer
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Hint: Let $A \equiv ( - 6,2)$,$B \equiv (2, - 2)$, and $C \equiv ( - 2, - 5)$, and let P be the midpoint of AB. Use the midpoint formula \[(\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2})\] to find the coordinates of point P. Then use the distance formula \[d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} \] to compute the length of CP to get the answer.


Complete step by step solution:
We are given a triangle whose vertices are $( - 6,2)$,$(2, - 2)$, and$( - 2, - 5)$
A median of the triangle is passing through the vertex $( - 2, - 5)$
We are asked to compute the length of this median.
All we need is the other endpoint of the median.
Then we will be able to find the length using the distance formula:
The distance between two points (x1, y1) and (x2, y2) is denoted by d and is given by the formula
\[d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} \]
Let $A \equiv ( - 6,2)$,$B \equiv (2, - 2)$, and $C \equiv ( - 2, - 5)$.
Then the median passing through point C will bisect side AB.
Therefore, let P be the midpoint of AB.
This implies that length of CP is the required length of the median.
The diagrammatic representation of the given information will be as follows:
seo images

For finding the coordinates of P, we use the midpoint formula for two points.
The midpoint of a segment formed by joining two points $({x_1},{y_1})$ and $({x_2},{y_2})$ is given by \[(\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2})\].
Using this formula, we have
Midpoint P of side AB
$
   \equiv (\dfrac{{ - 6 + 2}}{2},\dfrac{{2 + ( - 2)}}{2}) \\
   \equiv ( - 2,0) \\
 $
Now, using the coordinates of the points C and P, and the distance formula, we can compute the length of CP as follows
\[d = \sqrt {{{( - 2 - ( - 2))}^2} + {{(0 - {{( - 5)}_1})}^2}} = \sqrt {25} = 5\]
Hence, the length of the median is 5 units.


Note: First draw the figure using the given data so that you will get a clear idea of how to proceed with the problem.Do not continue without figure as there are chances of going wrong