Question

COLUMN A: The perimeter of the shaded region in the rectangle
COLUMN B: \(2\sqrt{2}+2\)

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

When you see a complex shape, look for ways to break it down into simpler right-angled triangles. The Pythagorean theorem is one of the most useful tools in geometry for finding unknown side lengths.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to find the perimeter of the shaded triangle inside the rectangle. The perimeter is the sum of the lengths of its three sides.
Step 2: Key Formula or Approach:
1. Use the dimensions of the rectangle to determine the base and height of the triangle. 2. The triangle is isosceles. We can use the Pythagorean theorem to find the length of the two equal slanted sides. The theorem states \(a^2 + b^2 = c^2\). 3. The perimeter is the sum of the three side lengths.
Step 3: Detailed Explanation:
From the diagram: The rectangle has a width of 2 and a height of 1. The shaded region is a triangle. The base of the triangle is the same as the width of the rectangle, so base = 2. The triangle is isosceles, and its height is the same as the height of the rectangle, which is 1. The height of the isosceles triangle bisects the base into two segments of length 1 each. This creates two identical right-angled triangles, each with a base of 1 and a height of 1. Let the length of the slanted side be \(c\). Using the Pythagorean theorem: \[ 1^2 + 1^2 = c^2 \] \[ 1 + 1 = c^2 \] \[ c^2 = 2 \] \[ c = \sqrt{2} \] The shaded triangle has three sides: - The base of length 2. - Two equal slanted sides, each of length \( \sqrt{2} \). Now, calculate the perimeter of the shaded triangle: \[ \text{Perimeter} = \text{base} + \text{side}_1 + \text{side}_2 \] \[ \text{Perimeter} = 2 + \sqrt{2} + \sqrt{2} = 2 + 2\sqrt{2} \] Comparison: Column A: The perimeter is \(2\sqrt{2} + 2\). Column B: The quantity is \(2\sqrt{2} + 2\). The two quantities are equal.
Step 4: Final Answer:
The perimeter of the shaded triangle is \(2 + 2\sqrt{2}\), which is equal to the quantity in Column B.