Question

The base of an equilateral triangle is
along the line given by 3x + 4y =9. If a
vertex of the triangle is (1, 2), then the
length of a side of the triangle is :

Answer 1

Answer:

The length of each side of the triangle is 4√3 / 15.

Step-by-step explanation:

Given data:

Let the base BC of an equilateral ∆ ABC be along the line given by 3x + 4y = 9

Coordinates of Vertex A of the triangle is (1,2)

To find: length of a side of the triangle.

Considering the length of each side of equilateral triangle be “a”.

Let us draw a perpendicular from A to BC at D such that BD = DC = a/2  

Therefore,  by Pythagoras theorem,

Length of AD = √[AB² - BD²] = √[a²-(a/2)² = [√3] a / 2 …. (i)

Also,

The shortest distance, AD of A(1,2) from 3x+4y=9 is given by

AD = | [ax1+by1+c] / [√[a²-b²] |

Or, AD = | [(3*1)+(4*2)-9] / [√[3²+4²] |

Or, AD = | 2/5 | ….. (ii)

Thus, equating (i) & (ii), we get

[√3] a / 2 = 2/5

Or, a = 4/(5*√3) = [4/(5*√3)] * [√3/√3] = 4√3 / 15