Question

The equation of a side of an equilateral triangle is 2x+y=3 and the coordinates of the opposite vertex are (1, –2), the area of the triangle (in sq. units) is

Answer 1

Given :-

• Equation of a side of an equilateral triangle is 2x+y = 3.
• coordinates of the opposite vertex are (1, –2).

To Find :-

• The area of the triangle (in sq. units) is ?

Formula used :-

• Length of Perpendicular distance from line Ax + By + c = 0 and (x1, y1) is :- | (Ax1 + By1 + C) / √(A² + B²) |
• sin 60° = (Perpendicular / Hypotenuse) = (√3/2) .
• Area of Equaliteral ∆ = (√3/4) * (side)²
• Each angle of an Equaliteral ∆ is 60° .

Solution :-

First we will drop a perpendicular from the opposite vertex (1, -2) to the line 2x + y - 3 = 0 .

So,

→ Length of Perpendicular = | (2*1 - 1*2 - 3) / √(2² + 1²) |

→ Length of Perpendicular = | (-3) / √5 |

→ Length of Perpendicular = (3/√5)

Now,

In Right angled ∆ made by Perpendicular ,

→ sin60° = (Perpendicular / Hypotenuse)

→ (√3/2) = (Perpendicular / side of Equaliteral ∆)

→ (√3/2) = (3 /√5 * side of Equaliteral ∆)

→ 6 = √15 * side of Equaliteral ∆

→ side of Equaliteral ∆ = (6/√15)

→ side of Equaliteral ∆ = (6√15/15)

→ side of Equaliteral ∆ = (2√15/5)

Therefore,

→ Area of Equaliteral ∆ = (√3/4) * (side)²

→ Area = (√3/4) * (2√15/5)²

→ Area = (√3/4) * (4 * 15/25)

→ Area = (√3 * 15) / 25

→ Area = (√3 * 3) / 5

→ Area = (3√3/5) unit². (Ans.)