Exercise 4 Page 342

To find the value of $x$ in the given diagram, let's start by labeling the vertices of each of the smaller triangles.

Looking at the markings, we can note a couple key congruence relationships.

\overline{A X} ≅ \overline{X B} \overline{C Y} ≅ \overline{Y B}

This means that

X

and

Y

are the midpoints of

\overline{A B}

and

\overline{C B},

respectively. Therefore,

\overline{X Y}

△ A B C .

The Triangle Midsegment Theorem tells us that, if a segment joins the midpoints of two sides of a triangle, then the segment is

half

as long as the third side.

X Y = \frac{1}{2} A C

Finally, we will substitute the given values into this equation to find

x .

x = \frac{1}{2} (30) \Leftrightarrow x = 15