The problem states that in the given diagram, $P$ and $Q$ are the midpoints of $YZ$ and $XZ$ respectively. You are asked to show that:

$4 \times \text{Area}(\triangle PQZ) = \text{Area}(\triangle XYZ)$

Let's go through the steps to prove this statement.

Step 1: Understand the Geometry

1. Midpoints: Since $P$ and $Q$ are midpoints:

   • $P$ divides $YZ$ into two equal parts.
   • $Q$ divides $XZ$ into two equal parts.

2. Triangles:

   • $\triangle PQZ$ is a smaller triangle formed within $\triangle XYZ$.

Step 2: Relationship Between the Areas of Similar Triangles

When two triangles are similar, the ratio of their areas is the square of the ratio of their corresponding sides.

• $\triangle PQZ$ and $\triangle XYZ$ are not necessarily similar, but $\triangle PQZ$ is formed by joining midpoints of $\triangle XYZ$.

Step 3: Apply the Midpoint Theorem

The midpoint theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half as long.

• In $\triangle XYZ$, the line $PQ$ (joining midpoints $P$ and $Q$) is parallel to $XY$ and half its length.

Step 4: Calculate Area Ratios

• Because $PQ \parallel XY$ and $PQ = \frac{1}{2} XY$, the height of $\triangle PQZ$ with respect to the base $PQ$ is also half the height of $\triangle XYZ$ with respect to the base $XY$.

• Therefore, the area of $\triangle PQZ$ is $\frac{1}{4}$ of the area of $\triangle XYZ$ because both the base and height are halved.

Step 5: Conclude the Proof

Since the area of $\triangle PQZ$ is $\frac{1}{4}$ of the area of $\triangle XYZ$:

$4 \times \text{Area}(\triangle PQZ) = \text{Area}(\triangle XYZ)$

This completes the proof.

Do you want more details on any of these steps or have any questions?

Here are 5 related questions for further exploration:

1. What is the Midpoint Theorem and how is it used in geometry?
2. How does the area ratio change if $P$ and $Q$ are not midpoints?
3. Can you derive the area formula for a triangle given its vertices in coordinate geometry?
4. How would the proof change if the triangle was not isosceles?
5. How can you generalize this proof to any triangle and any internal point dividing the sides proportionally?

Tip: Always check if smaller triangles within a larger triangle share proportional relationships, as this often simplifies area calculations.