Question

Quadrilateral ABCD is a parallelogram. Name the principal theorem or definition that justifies the statement.

$\mathrm{AX}=\frac{1}{2}\mathrm{AC}$

Short answer

The theorem that justifies the statement $AX=\frac{1}{2}AC$is theorem 5-3 which states that diagonals of a parallelogram bisect each other.

Step-by-step solution

Step 1. Observe the diagram.

The given diagram is:

Step 2. Description of step.

The theorem 5-3 states that diagonals of a parallelogram bisect each other.

It is being given that quadrilateral ABCD is a parallelogram.

From the given diagram it can be noticed that $\overline{AC}$ and$\overline{BD}$ are the diagonals of the parallelogram ABCD.

Therefore, by using the theorem 5-3 it can be said that the diagonals $\overline{AC}$and $\overline{BD}$will bisect each other.

From the given diagram it can be noticed that the diagonals $\overline{AC}$ and $\overline{BD}$ are bisecting each other at X.

Therefore, X is the midpoint of $\overline{AC}$.

Therefore, by using the definition of midpoint it can be said that:

$AX=XC$

From the given diagram it can be noticed that:

$AX+XC=AC$

Therefore, it can be obtained that:

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

Therefore,$AX=\frac{1}{2}AC$

Step 3. Write the theorem that justifies the statement.

Therefore, the theorem that justifies the statement $AX=\frac{1}{2}AC$is theorem 5-3 which states that diagonals of a parallelogram bisect each other.