Chapter 5.1, Problem 13CE

To determine

To state: the theorem or definition which justifies the given statement, “ $AX=\frac{1}{2}AC$ ”

Answer to Problem 13CE

Theorem 5-3

Explanation of Solution

Given:

The diagram:

  $ABCD$ is a parallelogram.

Concept Used:

Theorem 5-3: Diagonals of a parallelogram bisect each other.

Consider the given figure:

It is given that $ABCD$ is a parallelogram.

So, by theorem 5-3, $\overline{AC}$ and $\overline{BD}$ bisect each other.

That is, C is the midpoint of $\overline{AC}$ and $\overline{BD}$ .

Thus,

  $AX=XC$

Now consider,

  $\begin{array}{c}AX+XC=AC\\ AX+AX=AC\\ 2AX=AC\\ AX=\frac{1}{2}AC\end{array}$