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Question
Find the length of the side and perimeter of an equilateral triangle whose height is `sqrt3` cm.
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Solution
Let ΔABC be given equilateral triangle and seg AD be its height.
AD = `sqrt3` cm
∠B = 60° ...(1) (Angle of an equilateral triangle)
In ΔADB,
∠ADB + ∠ABD + ∠BAD = 180° ...(Sum of all angles of a triangle is 180°)
∴ 90° + 60° + ∠BAD = 180°
∴ 150° + ∠BAD = 180°
∴ ∠BAD = 180° − 150°
∴ ∠BAD = 30°
∴ ΔADB is a 30° - 60° - 90° triangle.
∴ by 30° - 60° - 90° triangle theorem.
AD = `sqrt3/2` AB ...(Side opposite to 60°)
∴ `sqrt3 = sqrt3/2` AB
∴ AB = `(2 xx sqrt3)/sqrt3`
∴ AB = 2 cm
∴ side of an equilateral triangle is 2 cm.
Perimeter of ΔABC = 3 × side
= 3 × AB
= 3 × 2
= 6 cm
∴ Perimeter of equilateral triangle is 6 cm.
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