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As shown in figure, P Q is a quarter circle with center R. □ABCR is a rectangle with A on
P R, C on RQ and B on the circle. OB = 4, OQ = 8 where O is center of rectangle. Find area
of rectangle ABCR.?
P R, C on RQ and B on the circle. OB = 4, OQ = 8 where O is center of rectangle. Find area
of rectangle ABCR.?
Most Upvoted Answer
As shown in figure, P Q is a quarter circle with center R. □ABCR is a ...
Understanding the Geometry
To analyze the problem, we start by identifying the key components of the geometric figure.
- Quarter Circle: P Q is a quarter circle with center R.
- Rectangle: ABCD is a rectangle inscribed within the quarter circle.
Given Data
- OB (Distance from center O to point B on the circle) = 4 units
- OQ (Distance from center O to point Q) = 8 units
Finding Dimensions of the Rectangle
- Radius of Quarter Circle: Since the quarter circle has a radius of 8 units (OQ), the rectangle's top side (BC) will be parallel to the x-axis, and the height (AB) will be vertical.
- Height of Rectangle: The height from O to B is given as 4 units, which implies:
- OB = 4 (height)
- Width of Rectangle: The base width (AC) can be calculated using the relation of circle radius:
- Since OB is vertical and the radius is horizontal, we determine that:
- OA (half of the base) = √(radius^2 - OB^2)
- Calculating OA:
- OA = √(8^2 - 4^2) = √(64 - 16) = √48 = 4√3
- Total Width (AC): Since OA is half, total width = 2 * OA = 8√3
Calculating Area of Rectangle
- Area Formula: Area = Height * Width
- Area = 4 * 8√3 = 32√3 square units
Conclusion
The area of rectangle ABCR is 32√3 square units. This represents the space enclosed within the rectangle inscribed in the quarter circle.
To analyze the problem, we start by identifying the key components of the geometric figure.
- Quarter Circle: P Q is a quarter circle with center R.
- Rectangle: ABCD is a rectangle inscribed within the quarter circle.
Given Data
- OB (Distance from center O to point B on the circle) = 4 units
- OQ (Distance from center O to point Q) = 8 units
Finding Dimensions of the Rectangle
- Radius of Quarter Circle: Since the quarter circle has a radius of 8 units (OQ), the rectangle's top side (BC) will be parallel to the x-axis, and the height (AB) will be vertical.
- Height of Rectangle: The height from O to B is given as 4 units, which implies:
- OB = 4 (height)
- Width of Rectangle: The base width (AC) can be calculated using the relation of circle radius:
- Since OB is vertical and the radius is horizontal, we determine that:
- OA (half of the base) = √(radius^2 - OB^2)
- Calculating OA:
- OA = √(8^2 - 4^2) = √(64 - 16) = √48 = 4√3
- Total Width (AC): Since OA is half, total width = 2 * OA = 8√3
Calculating Area of Rectangle
- Area Formula: Area = Height * Width
- Area = 4 * 8√3 = 32√3 square units
Conclusion
The area of rectangle ABCR is 32√3 square units. This represents the space enclosed within the rectangle inscribed in the quarter circle.
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As shown in figure, P Q is a quarter circle with center R. □ABCR is a rectangle with A on P R, C on RQ and B on the circle. OB = 4, OQ = 8 where O is center of rectangle. Find area of rectangle ABCR.?
Question Description
As shown in figure, P Q is a quarter circle with center R. □ABCR is a rectangle with A on P R, C on RQ and B on the circle. OB = 4, OQ = 8 where O is center of rectangle. Find area of rectangle ABCR.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about As shown in figure, P Q is a quarter circle with center R. □ABCR is a rectangle with A on P R, C on RQ and B on the circle. OB = 4, OQ = 8 where O is center of rectangle. Find area of rectangle ABCR.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for As shown in figure, P Q is a quarter circle with center R. □ABCR is a rectangle with A on P R, C on RQ and B on the circle. OB = 4, OQ = 8 where O is center of rectangle. Find area of rectangle ABCR.?.
As shown in figure, P Q is a quarter circle with center R. □ABCR is a rectangle with A on P R, C on RQ and B on the circle. OB = 4, OQ = 8 where O is center of rectangle. Find area of rectangle ABCR.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about As shown in figure, P Q is a quarter circle with center R. □ABCR is a rectangle with A on P R, C on RQ and B on the circle. OB = 4, OQ = 8 where O is center of rectangle. Find area of rectangle ABCR.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for As shown in figure, P Q is a quarter circle with center R. □ABCR is a rectangle with A on P R, C on RQ and B on the circle. OB = 4, OQ = 8 where O is center of rectangle. Find area of rectangle ABCR.?.
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