Question 1209532: Point R is chosen on the circumference of a circle and arc PQ is drawn with R as its center. If P and Q are joined, the line segment formed is a diameter of the circle with length 2r cm. Find the area of the shaded region in terms of r, in square centimeters.
Link to diagram: https://ibb.co/GQFdy2SV
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to find the area of the shaded region:
1. **Area of the Semicircle:** Since PQ is the diameter, the radius of the circle is r. The area of the entire circle is πr², so the area of the semicircle (the region above PQ) is (1/2)πr².
2. **Area of Triangle PQR:** Triangle PQR is inscribed in a semicircle with PQ as the diameter. A fundamental theorem states that any triangle inscribed in a semicircle with the diameter as one side is a right triangle. Therefore, angle PRQ is a right angle.
* The base of triangle PQR is PQ (the diameter), which has a length of 2r.
* The height of triangle PQR is the perpendicular distance from R to PQ. Since R is on the circle's circumference, this height is equal to the radius, r.
* Area of triangle PQR = (1/2) * base * height = (1/2) * 2r * r = r²
3. **Area of the Shaded Region:** The shaded region is the area of the semicircle *minus* the area of triangle PQR.
Shaded Area = Area of Semicircle - Area of Triangle PQR
Shaded Area = (1/2)πr² - r²
4. **Simplify:** We can factor out r²:
Shaded Area = r² * [(π/2) - 1]
Therefore, the area of the shaded region is **r²[(π/2) - 1]** square centimeters.
Answer by ikleyn(52772)  (Show Source):
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