A rectangle R is inscribed in a circle of radius 6 cm. Which of the following statements is/are true? The maximum area of R is 72 sq cm. The least perimeter of R is 24√2 units.
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see figure, Let a rectangle is inscribed in a circle of radius r in such a way that diameter of circle equals diagonal of rectangle. and diameter of circle makes an angle ∅ with breadth of rectangle as shown in figure.
then, L = 2rsin∅ , B = 2rcos∅
so, area of rectangle = L × B
= 4r² sin∅.cos∅
= 2r²(2sin∅.cos∅)
we know, 2sinx.cosx = sin2x
so, Area of rectangle = 2r²sin2∅
area will be maximum only when sin2∅ will be maximum.
we know maximum value of sine function is 1
so, sin2∅ = 1 = sin90°
so, ∅ = 45°
hence, L = 2r sin45° = √2r
B = 2r cos45° = √2r
hence, maximum area of rectangle = (√2r)(√2r)
= 2r² = 2 × (6)² = 72 cm²
maximum perimeter of rectangle = 2(√2r + √2r) = 4√2 r
= 24√2 unit
hence, first one is correct e.g., maximum area of R is 72 sq cm²
then, L = 2rsin∅ , B = 2rcos∅
so, area of rectangle = L × B
= 4r² sin∅.cos∅
= 2r²(2sin∅.cos∅)
we know, 2sinx.cosx = sin2x
so, Area of rectangle = 2r²sin2∅
area will be maximum only when sin2∅ will be maximum.
we know maximum value of sine function is 1
so, sin2∅ = 1 = sin90°
so, ∅ = 45°
hence, L = 2r sin45° = √2r
B = 2r cos45° = √2r
hence, maximum area of rectangle = (√2r)(√2r)
= 2r² = 2 × (6)² = 72 cm²
maximum perimeter of rectangle = 2(√2r + √2r) = 4√2 r
= 24√2 unit
hence, first one is correct e.g., maximum area of R is 72 sq cm²
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