Home
Class 12
MATHS
In an equilateral triangle, three coins ...

In an equilateral triangle, three coins of radii 1 unit each are kept so that they touch each other and also the sides of the triangle. The area of the triangle is 2sqrt(3)` (b) `6+4sqrt(3)` `12+(7sqrt(3))/4` (d) `3+(7sqrt(3))/4`

Text Solution

Verified by Experts

The correct Answer is:
`(4sqrt3+6)` sq. unit
The given arrangement of coins is as shown in the given figure.

To find the area of the triangle, we need to find its side.
For circle with centre `C_2`, BP and BP' are two tangents from B to circle,
Therefore, `BC_(2)` must be angle bisector of `angleB`.
`angle B=60^@" "( :.DeltaABC" is equilateral")`
`:. angleC_(2)BP=30^(@)`
`:. tan30^(@)=1/x`
`rArr x=sqrt3`
`:. BC=BP+PQ+QC`
`=x+2+x=2+2sqrt3`
Area of an equilateral triangle is `sqrt3/4 a^2`, where a is side length.
`:. "Area of " DeltaABC=sqrt3/4(2+2sqrt3)^2`
`=sqrt3(1+3+2sqrt3)`
`=(4sqrt3+6)` sq. unit
Promotional Banner

Topper's Solved these Questions

  • TRIGONOMETRIC FUNCTIONS

    CENGAGE PUBLICATION|Exercise Concept Application Exercises 2.2|8 Videos

  • TRIGONOMETRIC FUNCTIONS

    CENGAGE PUBLICATION|Exercise Concept Application Exercises 2.3|4 Videos

  • TRIGONOMETRIC FUNCTIONS

    CENGAGE PUBLICATION|Exercise example|10 Videos

  • TRIGONOMETRIC EQUATIONS

    CENGAGE PUBLICATION|Exercise Archives (Numerical value type)|4 Videos

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE PUBLICATION|Exercise Archives (Numerical Value Type)|2 Videos

Similar Questions

Explore conceptually related problems

If the ratio of the length of the sides of a triangle be 1:sqrt3:2 then the ratio of the angle be

Find the angles of the triangle whose sides are (sqrt(3)+1)/(2sqrt(2)), (sqrt(3)-1)/(2sqrt(2)) and sqrt(3)/2 .

Find the angles of the triangle whose sides are y+2=0,sqrt(3)x+y=1andy-sqrt(3)x+2+3sqrt(3)=0

The area of an equilateral triangle is 4sqrt(3)cm^(2). Then the of each of the sides of the triangle is

If the side of an equilateral triangle increases at the rate of sqrt(3) cm/s and its area at the rate of 12 cm^(2) /s, find the side of the triangle .

Find the area of the triangle with vertices at (-4,1),(1,2),(4,-3).

On the line segment joining (1, 0) and (3, 0) , an equilateral triangle is drawn having its vertex in the fourth quadrant. Then the radical center of the circles described on its sides. (a) (3,-1/(sqrt(3))) (b) (3,-sqrt(3)) (c) (2,-1/sqrt(3)) (d) (2,-sqrt(3))

One angle of an isosceles triangle is 120^0 and the radius of its incircle is sqrt(3)dot Then the area of the triangle in sq. units is (a) 7+12sqrt(3) (b) 12-7sqrt(3) (c) 12+7sqrt(3) (d) 4pi

A rectangle is inscribed in an equilateral triangle of side length 2a units. The maximum area of this rectangle can be (a) sqrt(3)a^2 (b) (sqrt(3)a^2)/4 a^2 (d) (sqrt(3)a^2)/2

The incentre of a triangle with vertices (7, 1),(-1, 5) and (3+2sqrt(3),3+4sqrt(3)) is