Q37. - Algebra 2

Question

Triangle CAT has vertices$C\left(4,9\right)$, A\left( 8,-9 \right)$and$T\left( -6,5 \right). M is the midpoint of \overline{TA}. Find the length of median$\overline{CM}$. (Hint: A median connects a vertex of a triangle to the midpoint of the opposite side.)

 

Step-by-Step Solution

Answer

The length of the median $\overline{CM}$ is 11.4 units.

Step 1. Given Information.

Given triangle CAT has vertices $C\left(4,9\right)$, $A\left(8,-9\right)$ and $T\left(-6,5\right)$. M is the midpoint of $\overline{TA}$.

The length of the median $\overline{CM}$ is to be determined. 

 

Step 2. Explanation .

The midpoint of two points $\left(x_1,y_1\right),\left(x_2,y_2\right)$ is given by $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

Plugging the values in the equation to find the point M:

$\begin{array}{l}M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ M=\left(\frac{8+\left(-6\right)}{2},\frac{\left(-9\right)+5}{2}\right)\\ M=\left(\frac{8-6}{2},\frac{-9+5}{2}\right)\\ M=\left(\frac{2}{2},\frac{-4}{2}\right)\\ M=\left(1,-2\right)\end{array}$

 

The distance between two points$\left(x_1,y_1\right),\left(x_2,y_2\right)$ is given by $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

 

Plugging the given values in the equation to find the distance between C and M:

$\begin{array}{l}d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ d=\sqrt{{\left(1-4\right)}^2+{\left(-2-9\right)}^2}\\ d=\sqrt{{\left(-3\right)}^2+{\left(-11\right)}^2}\\ d=\sqrt{9+121}\\ d=\sqrt{130}\\ d=11.4\text{ units}\end{array}$

Step 3. Conclusion .

Hence, the length of the median $\overline{CM}$ is 11.4 units.