Question

In Exercises 16-17 a segment joins the midpoints of two sides of a triangle.

Find the values of x and y.

Short answer

The length of x and y is 4 units and 2 units respectively.

Step-by-step solution

Step 1. Consider the diagram.

The triangle is shown below:

The line segment joins the midpoints of the triangle sides. Length of sides are $3x-2y$ and 8. Length of midsegment and base of the triangle are 7 and$5x-3y$ respectively.

Step 2. Show the calculation.

Line segment passes through the midpoint of the side of the triangle.

Hence,

$3x-2y=8$

That implies,

$x=\frac{\left(8+2y\right)}{3}$ …… (1)

According to the midsegment theorem, the length of the midsegment is half of the base of the triangle.

Hence,

$\begin{array}{c}7=\frac{1}{2}\left(5x-3y\right)\\ 14=5x-3y\\ 5x=14+3y\end{array}$

From equation (1), $x=\frac{\left(8+2y\right)}{3}$, then,

$\begin{array}{c}5x=14+3y\\ 5\left(\frac{\left(8+2y\right)}{3}\right)=14+3y\\ 40+10y=42+9y\\ y=2\text{units}\end{array}$

Substitute 2 for y in equation (1) as follows:

$\begin{array}{c}x=\frac{\left(8+2y\right)}{3}\\ =\frac{\left(8+2\left(2\right)\right)}{3}\\ =4\text{units}\end{array}$

Thus, the length of x is 4 units and y is 2 units.

Step 3. State the conclusion.

Therefore, the length of x and y is 4 units and 2 units respectively.