Question

In the image shown below, PQ = QR = QS = 2 units. Also, QT = TS = QU = UR. The points T and U are the centre points of the semicircles. What is the total area of shaded portions in square units?

Hint:

To calculate the area of semicircles, use the formula \(\frac{1}{2} \pi r^2\), where \(r\) is the radius.

Answer 1

Correct Answer: 2

Step 1: Analyze the given dimensions.
We are given that PQ = QR = QS = 2 units. Points T and U are the centers of the semicircles on the sides of triangle PQR. Since the shaded regions consist of areas from the semicircles, we need to calculate the area of those shaded portions.
Step 2: Area of shaded regions.
Each semicircle has a radius of 1 unit (half of 2 units). The area of a semicircle is given by: \[ \text{Area of semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (1^2) = \frac{\pi}{2} \] The total area of shaded regions (all semicircles) is the sum of the areas of the two semicircles.
Step 3: Final Calculation.
Thus, the total area of shaded portions is: \[ \boxed{2} \, \text{square units.} \]