Mensuration
- If the numerical values of the height and the area of an equilateral triangle be same, then the length of each side of the triangle is
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AB = 2x units
BD = DC = x units
AD = √AB² - BD²
= √4x² - x²
= √3x²
= √3x unitsArea of ∆ABC = √3 × (2x)² 4
According to question,∵ √3 × (2x)² = √3x 4 ⇒ √3 × 4x² = √3x 4
⇒ x² = x ⇒ x (x – 1) = 0
⇒ x = 1 Hence length of side 2 × 1 = 2 units
∴ Length of side = 9 unitsCorrect Option: A
AB = 2x units
BD = DC = x units
AD = √AB² - BD²
= √4x² - x²
= √3x²
= √3x unitsArea of ∆ABC = √3 × (2x)² 4
According to question,∵ √3 × (2x)² = √3x 4 ⇒ √3 × 4x² = √3x 4
⇒ x² = x ⇒ x (x – 1) = 0
⇒ x = 1 Hence length of side 2 × 1 = 2 units
∴ Length of side = 9 units
- If the length of a side of the square is equal to that of the diameter of a circle, then the ratio of the area of the square and that of the circle is ( π = 22/7)
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Length of side of square = 2x units
Diameter of circle = 2x units
Radius = x units
∴ Required ratio = 4x² : πx²= 4 : 22 7
= 14 : 11Correct Option: A
Length of side of square = 2x units
Diameter of circle = 2x units
Radius = x units
∴ Required ratio = 4x² : πx²= 4 : 22 7
= 14 : 11
- The median of an equilateral triangle is 6√3 cm. The area (in cm2) of the triangle is
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Using Rule 6,
AB = BC = AC = a cm AD = Median = 6√3 cm.∴ √3 a = 6√3 2 ⇒ a = 6√3 × 2 = 12 cm. √3 ∴ Area of ∆ABC = √3 × side² = √3 × 12 × 12 4 4
= 36√3 sq. cm.Correct Option: D
Using Rule 6,
AB = BC = AC = a cm AD = Median = 6√3 cm.∴ √3 a = 6√3 2 ⇒ a = 6√3 × 2 = 12 cm. √3 ∴ Area of ∆ABC = √3 × side² = √3 × 12 × 12 4 4
= 36√3 sq. cm.
- If the numerical value of the circumference and area of a circle is same, then the area is
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Using Rule 14,
Radius of circle = r units
According to question,
Area of circle = circumference of circle
⇒ πr² = 2πr
⇒ r = 2 units
∴ Area of circle = πr²
= 4π sq. unitsCorrect Option: B
Using Rule 14,
Radius of circle = r units
According to question,
Area of circle = circumference of circle
⇒ πr² = 2πr
⇒ r = 2 units
∴ Area of circle = πr²
= 4π sq. units
- The area of an equilateral triangle is 48 sq. cm. The length of the side is
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Using Rule 6,
Area of equilateral triangle = √3 × side² 4 ⇒ √3 × (side)² = 48 4 (side)² = 48 × 4 √3 = 16 × √3 × √3 × 4 = 64√3 √3
∴ Side = √64√3 = 84√3cmCorrect Option: E
Using Rule 6,
Area of equilateral triangle = √3 × side² 4 ⇒ √3 × (side)² = 48 4 (side)² = 48 × 4 √3 = 16 × √3 × √3 × 4 = 64√3 √3
∴ Side = √64√3 = 84√3cm
