Question

Let XYZ be an equilateral triangle and let P, Q, R be the midpoints of YZ, XZ, and XY, respectively.
Let \( r = \frac{\text{Area of } \Delta PQR}{\text{Area of } \Delta XYZ} \).

Hint:

When midpoints of the sides of a triangle are joined, the area of the resulting triangle is one-quarter of the area of the original triangle.

Answer 1

Correct Answer: 0.25

Step 1: Understanding the area ratio.
In any triangle, the area of the triangle formed by joining the midpoints of the sides is one-quarter of the area of the original triangle. This is because the midpoints divide the sides of the triangle into smaller, proportional triangles, and their areas are related by a factor of 1/4.
Step 2: Using the formula.
We are given the formula: \[ r = \frac{\text{Area of } \Delta PQR}{\text{Area of } \Delta XYZ} \] Since \( \Delta PQR \) is formed by the midpoints, its area is one-quarter of the area of \( \Delta XYZ \). Hence, \( r = \frac{1}{4} \).
Step 3: Conclusion.
The value of \( r \) is 1/4.