Question

If the area of the shaded region of the square above is 20, what is the perimeter of the square? 

 

• A. \(4\sqrt{5}\)
• B. \(8\sqrt{5}\)
• C. \(16\sqrt{5}\)
• D. 80
• E. 400

Hint:

If a diagram in a geometry problem is ambiguous, consider different plausible interpretations. The correct one is usually the one that leads to one of the given answer choices. Here, interpreting the triangle's area as \(\frac{s^2}{2}\) (base=side, height=side) leads to an answer not in the options, while interpreting it as \(\frac{s^2}{4}\) (base=side, height=half side) leads to a correct option.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires finding the perimeter of a square given the area of a triangular region within it. The key is to correctly interpret the geometry of the triangle from the diagram to relate its area to the side length of the square. While the diagram is not perfectly clear, we can deduce the intended geometry by finding an interpretation that leads to one of the answer choices. A likely interpretation is that the triangle's vertices are one corner of the square and the midpoints of the two opposite sides.
Step 2: Key Formula or Approach:
1. Let the side length of the square be \(s\).
2. Express the area of the shaded triangle in terms of \(s\). The area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\).
3. Set this area expression equal to 20 and solve for \(s\).
4. Calculate the perimeter using the formula \(P = 4s\).
Step 3: Detailed Explanation:
Let the side length of the square be \(s\). Let's assume the vertices of the shaded triangle are one corner (e.g., the bottom-left) and the midpoints of the two opposite sides (top and right sides).
Let's place the square on a coordinate plane with vertices at (0,0), (s,0), (s,s), and (0,s).
The triangle's vertices would be at (0,0), the midpoint of the top side \((s/2, s)\), and the midpoint of the right side \((s, s/2)\).
The area of this triangle can be found by taking the area of the square and subtracting the three unshaded right triangles in the corners.
- Triangle 1 (bottom right): base \(s/2\), height \(s\). Area = \(\frac{1}{2}(s/2)(s) = s^2/4\). No, base s, height s/2. Area = \(\frac{1}{2}(s)(s/2) = s^2/4\).
- Triangle 2 (top left): base \(s/2\), height \(s\). Area = \(\frac{1}{2}(s/2)(s) = s^2/4\).
- Triangle 3 (top right): base \(s/2\), height \(s/2\). Area = \(\frac{1}{2}(s/2)(s/2) = s^2/8\).
This is too complicated and depends on a specific interpretation.
Let's try a simpler interpretation that fits the options. Let the vertices of the triangle be the midpoint of the top side, the midpoint of the bottom side, and one of the other corners (e.g., the bottom-left).
- Base of the triangle: The line connecting the midpoints \((s/2, 0)\) and \((s/2, s)\). This is a vertical line of length \(s\).
- Height of the triangle: The perpendicular distance from the corner vertex (0,0) to the base line \(x=s/2\). This distance is \(s/2\).
- Area = \(\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times s \times \frac{s}{2} = \frac{s^2}{4}\).
This interpretation gives a simple formula. Let's assume it is the correct one.
We are given that the area of the shaded region is 20.
\[ \frac{s^2}{4} = 20 \] Multiply both sides by 4 to solve for \(s^2\):
\[ s^2 = 80 \] Take the square root of both sides to find the side length \(s\):
\[ s = \sqrt{80} \] To simplify the square root, find the largest perfect square factor of 80. \(80 = 16 \times 5\).
\[ s = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5} \] Now, calculate the perimeter of the square:
\[ P = 4s = 4 \times (4\sqrt{5}) = 16\sqrt{5} \] Step 4: Final Answer:
The perimeter of the square is \(16\sqrt{5}\).