Question

In the figure above, the areas of square regions X and Y are 1 and 4, respectively. What is the area of the triangular region?

• A. 2
• B. 1
• C. \( \frac{3}{4} \)
• D. \( \frac{1}{2} \)
• E. \( \frac{1}{4} \)

Hint:

When a geometric figure is composed of simpler shapes, break it down. Use the properties of the known shapes (squares) to find the dimensions (base and height) of the shape you need to measure (triangle).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The problem asks for the area of a shaded triangle formed by the corners of two squares. We need to use the given areas of the squares to find the dimensions necessary to calculate the area of the triangle.
Step 2: Key Formula or Approach:
1. The area of a square is given by \( \text{Area} = \text{side}^2 \). We can find the side length by taking the square root of the area.
2. The area of a triangle is given by \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
Step 3: Detailed Explanation:
First, let's find the side lengths of the squares X and Y.
For square X: \[ \text{Area}_X = 1 \] \[ \text{side}_X = \sqrt{1} = 1 \] For square Y: \[ \text{Area}_Y = 4 \] \[ \text{side}_Y = \sqrt{4} = 2 \] Now, let's look at the shaded triangular region. It's a right-angled triangle.
The height of the triangle is the side length of the smaller square, square X. So, \( \text{height} = \text{side}_X = 1 \).
The base of the triangle is the difference between the side length of the larger square (Y) and the smaller square (X). \[ \text{base} = \text{side}_Y - \text{side}_X = 2 - 1 = 1 \] Now we can calculate the area of the triangle: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \] \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \] Step 4: Final Answer:
The area of the triangular region is \( \frac{1}{2} \).