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 3 units and 3 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 $4y+2$and $7\left(y-1\right)$. Length of midsegment and base of the triangle are $3x+5$ and $12x-8$ respectively.

Step 2. Show the calculation.

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

Hence,

$\begin{array}{l}4y+2=7\left(y-1\right)\\ 4y+2=7y-7\\ 3y=9\\ y=3\text{units}\end{array}$

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

Hence,

$\begin{array}{l}3x+5=\frac{1}{2}\left(12x-8\right)\\ 3x+5=6x-4\\ 3x=9\\ x=3\text{units}\end{array}$

Thus, the length of x is 3 units and y is 3 units.

Step 3. State the conclusion.

Therefore, the length of x and y is 3 units and 3 units respectively.