Question

Point$D\left(5,-1\right)$is the midpoint of segment $\overline{CE}$. If point Chas coordinates $\left(3,2\right)$then what are the coordinates of point E.

$\left(A\right)\left(8,1\right)\left(B\right)\left(7,-4\right)\left(C\right)\left(2,-3\right)\left(D\right)\left(4,\frac{1}{2}\right)$

Short answer

The coordinates of point E is$\left(7,-4\right)$. So, option (B) is correct.

Step-by-step solution

Step 1. Write down the given information.

The given segment $\overline{CE}$has midpoint $D\left(5,-1\right)$ and the point C has coordinates $\left(3,2\right)$$\left(3,2\right)$.

Step 2. Concept used.

If a line segment has end-points at$\left(x_1,y_1\right)\text{and}\left(x_2,y_2\right)$ then the midpoint of the segment has coordinates $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

Step 3. Calculation.

The coordinates of point E say $\left(h,k\right)$can be evaluated by using the midpoint formula.

$\begin{array}{c}\text{Midpointof}\overline{CE}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ \left(5,-1\right)=\left(\frac{3+h}{2},\frac{2+k}{2}\right)\\ 5=\frac{3+h}{2},-1=\frac{2+k}{2}\\ h+3=10,k+2=-2\\ h=7,k=-4\end{array}$

Since the point is$E\left(7,-4\right)$. So, option (B) is correct.

Step 4. Conclusion.

The coordinates of point E is $\left(7,-4\right)$. So, option (B) is correct.