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Find the maximum area of the isosceles trapezium (in units2) whose unequal sides are 4 units and 6 units. Given that it is inscribed in a circle whose radius is 
- a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Find the maximum area of the isosceles trapezium (in units2) whose une...
The maximum area of trapezium will be obtained when the two sides are on the opposite sides of the diameter.
The height of the trapezium = OE + OF
OE = √OC2 - √EC2 = √3
OF = √OB2 - √FB2 = 2√2
OE = √OC2 - √EC2 = √3
OF = √OB2 - √FB2 = 2√2
∴ the area of the trapezium = (1 / 2) x (2√2 + √3) x (6 + 4) = 5(2√2 + √3)
Hence, option 2.
Most Upvoted Answer
Find the maximum area of the isosceles trapezium (in units2) whose une...
The maximum area of trapezium will be obtained when the two sides are on the opposite sides of the diameter.
The height of the trapezium = OE + OF
OE = √OC2 - √EC2 = √3
OF = √OB2 - √FB2 = 2√2
OE = √OC2 - √EC2 = √3
OF = √OB2 - √FB2 = 2√2
∴ the area of the trapezium = (1 / 2) x (2√2 + √3) x (6 + 4) = 5(2√2 + √3)
Hence, option 2.
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Find the maximum area of the isosceles trapezium (in units2) whose une...
Understanding the Problem
To find the maximum area of an isosceles trapezium inscribed in a circle with radius 2√3, we need to analyze the geometric properties and apply the relevant formulas.
Given Parameters
- Unequal sides: 4 units and 6 units
- Radius of the circumcircle: 2√3
Properties of Isosceles Trapezium
- An isosceles trapezium has two parallel sides and two equal non-parallel sides.
- The area (A) can be calculated using the formula A = 1/2 * (a + b) * h, where a and b are the lengths of the parallel sides, and h is the height.
Using the Circumcircle
Since the trapezium is inscribed in a circle, we can derive the relationship between the sides and the radius.
- The trapezium is symmetrical, allowing us to drop perpendiculars from the endpoints of the shorter base to the longer base, forming two right triangles.
- By applying the Pythagorean theorem, we can find the height (h) of the trapezium in terms of the radius.
Maximizing the Area
- To maximize the area, we need to express both bases and the height in terms of the trapezium's sides and the radius.
- The area can be maximized by strategic positioning of the bases while ensuring the trapezium remains inscribed in the circle.
Final Calculation
After performing the necessary calculations, the maximum area of the isosceles trapezium can be expressed as:
- A = 5(2√2 + √3) square units.
Conclusion
The correct option for the maximum area of the isosceles trapezium is indeed option 'B': 5(2√2 + √3).
To find the maximum area of an isosceles trapezium inscribed in a circle with radius 2√3, we need to analyze the geometric properties and apply the relevant formulas.
Given Parameters
- Unequal sides: 4 units and 6 units
- Radius of the circumcircle: 2√3
Properties of Isosceles Trapezium
- An isosceles trapezium has two parallel sides and two equal non-parallel sides.
- The area (A) can be calculated using the formula A = 1/2 * (a + b) * h, where a and b are the lengths of the parallel sides, and h is the height.
Using the Circumcircle
Since the trapezium is inscribed in a circle, we can derive the relationship between the sides and the radius.
- The trapezium is symmetrical, allowing us to drop perpendiculars from the endpoints of the shorter base to the longer base, forming two right triangles.
- By applying the Pythagorean theorem, we can find the height (h) of the trapezium in terms of the radius.
Maximizing the Area
- To maximize the area, we need to express both bases and the height in terms of the trapezium's sides and the radius.
- The area can be maximized by strategic positioning of the bases while ensuring the trapezium remains inscribed in the circle.
Final Calculation
After performing the necessary calculations, the maximum area of the isosceles trapezium can be expressed as:
- A = 5(2√2 + √3) square units.
Conclusion
The correct option for the maximum area of the isosceles trapezium is indeed option 'B': 5(2√2 + √3).
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Find the maximum area of the isosceles trapezium (in units2) whose unequal sides are 4 units and 6 units. Given that it is inscribed in a circle whose radius isa)b)c)d)Correct answer is option 'B'. Can you explain this answer?
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