Question

EFGH is a parallelogram whose diagonals intersect at P. M is the midpoint of $\overline{\text{FG}}$. Prove that $\text{MP}=\frac{1}{2}\text{EF}$.

Short answer

$\text{MP}=\frac{1}{2}\text{EF}$

Step-by-step solution

Step 1. Consider the diagram.

EFGH is a parallelogram whose diagonals intersect at P. M is the midpoint of FG.

Join the line MP as shown below:

Step 2. Show the calculation.

From the diagram,

P is the diagonal of the parallelogram EFGH.

Hence, diagonals of a parallelogram bisect each other.

Thus, P is the midpoint of side EG. M is the midpoint of side FG.

According to the midpoint theorem,

$PM\parallel EF$

Hence,

$PM=\frac{1}{2}EF$

Step 3. State the conclusion.

Therefore, $PM=\frac{1}{2}EF$(proved).