Question

P, Q, R, S are the midpoints of the sides of square ABCD. What fraction of the square is the unshaded area?

• A. 2/3
• B. 1/2
• C. 3/4
• D. 5/6

Answer 1

Answer:

C. 3/4

Let's assume each of the 4 smaller squares has a side length of 1 unit.

• This makes the area of each small square = 1 ($1 \times 1$).

• The total area of the entire large square ABCD = 4 ($2 \times 2$).

Now, let's look closely at the top-left small square:

• The shaded triangle has a base along the top edge of length 1.

• The diagonal line goes from the top-left corner all the way to the right-middle point ($Q$). Because it slants downward across two squares, it hits the middle vertical line exactly halfway down.

• This means the height of that shaded triangle is only $\frac{1}{2}$.

• $\text{Area of this triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times \frac{1}{2} = \mathbf{\frac{1}{4}}$.

Totaling the Sections

Each of the 4 shaded triangles has the exact same base of 1 and height of $\frac{1}{2}$:

1. Top-Left triangle = $\frac{1}{4}$

2. Top-Right triangle = $\frac{1}{4}$

3. Bottom-Left triangle = $\frac{1}{4}$

4. Bottom-Right triangle = $\frac{1}{4}$

$\text{Total Shaded Area} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \mathbf{1 \text{ full square unit}}$

Since the total area of the entire figure is 4 square units:

• Shaded part = $1 \text{ out of } 4 = \mathbf{\frac{1}{4}}$

• Unshaded part = $3 \text{ out of } 4 = \mathbf{\frac{3}{4}}$