To determine the value of $r = \frac{\text{Area of triangle } Q}{\text{Area of square } P}$, we need to calculate the areas of both the square $P$ and the triangle $Q$.

1. Area of Square $P$:

   • Since $P$ is a square with side length 4 cm, its area is: $\text{Area of } P = 4 \times 4 = 16 \text{ cm}^2$

2. Area of Triangle $Q$:

   • The triangle $Q$ has a base of 4 cm and a height of 4 cm.
   • The area of a triangle is given by: $\text{Area of } Q = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 4 = \frac{1}{2} \times 16 = 8 \text{ cm}^2$

3. Finding $r$:

   • Now, we calculate $r$ by dividing the area of triangle $Q$ by the area of square $P$: $r = \frac{\text{Area of } Q}{\text{Area of } P} = \frac{8}{16} = \frac{1}{2}$

Thus, the value of $r$ is $\frac{1}{2}$.

The correct answer is therefore:

• $\frac{1}{2}$.

Would you like a detailed explanation or further clarification on this problem?

Here are some additional questions that build on this solution:

1. What would $r$ be if the height of the triangle $Q$ was doubled?
2. How would the area change if both the square and the triangle were scaled by a factor of 2?
3. If the side length of the square $P$ was increased by 1 cm, what would be the new value of $r$?
4. How is the formula for the area of a triangle derived?
5. Can the ratio $r$ be greater than 1 if both shapes share the same side length?

Tip: When working with areas of similar shapes, remember that the area scales by the square of the scaling factor.