Chapter 4.8, Problem 30HP

To determine

To calculate:

The coordinates of the other two verticesfrom three vertices given if one of the vertex is located at the origin.

Answer to Problem 30HP

The coordinates of the other two vertices are $\left(2a,0\right)$ and $\left(2a,2b\right)$ .

Explanation of Solution

Given information:

The midpoints of the sides of a triangle are located at $\left(a,0\right),\left(2a,b\right),\left(a,b\right)$ .

Calculation:

Let $A$ , $B$ and $C$ represent the vertices of the triangle with $A$ positioned at the origin.

Let $\left(a,0\right)$ be the midpoint of $\overline{AB}$ , $\left(2a,b\right)$ be the midpoint of $\overline{BC}$ , and $\left(a,b\right)$ be the midpoint of $\overline{AC}$ . Use the midpoint formula to determine thecoordinates of $B$ and $C$ .

  

By using the midpoint formula:

  $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Substitute the values,

  $\begin{array}{l}\left(a,0\right)=\left(\frac{0+x_2}{2},\frac{0+y_2}{2}\right)\\ 2a=x_{2}and0=y_2\end{array}$

Thus, the coordinates of point $B$ are $\left(2a,0\right)$ .

By using the midpoint formula:

  $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Substitute the values,

  $\begin{array}{l}\left(a,b\right)=\left(\frac{0+x_2}{2},\frac{0+y_2}{2}\right)\\ 2a=x_{2}and2b=y_2\end{array}$

Thus, the coordinates of point $C$ are $\left(2a,2b\right)$ .

Check by finding the midpoint of $\overline{BC}$ .

By using the midpoint formula:

  $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Substitute the values,

  $\begin{array}{l}{M}_{\overline{BC}}=\left(\frac{2a+2a}{2},\frac{0+2b}{2}\right)\\ M_{\overline{BC}}=\left(2a,b\right)\end{array}$

Thus, the coordinates of the other two vertices are $\left(2a,0\right)$ and $\left(2a,2b\right)$ .