Question

An equilateral triangle with one vertex (4.3) has
two other vertices on the line
3x – 4y – 20 = 0 and 3x – 4y + 10 = 0.
If the length of the side of the triangle is x, then
x2 is​

Answer 1

$\textbf{Given:}$

$\text{Two vertices of the given equilateral triangle are on the line}$

$\text{3x-4y-20=0 and 3x-4y+10=0}$

$\text{Length of the side of the triangle is x}$

$\textbf{To find:}$

$\text{Length of the side of the equilateral triangle}$

$\textbf{Solution:}$

$\text{It is clear that, the given two lines are parallel}$

$\text{Since two vertices of the given equilateral triangle are on the parallel lines}$

$\text{3x-4y-20=0 and 3x-4y+10=0, we have}$

$\text{Length of the side of the triangle}$

$\text{=Distance between the parallel lines}$

$=\displaystyle|\frac{c_1-c_2}{\sqrt{a^2+b^2}}|$

$=\displaystyle|\frac{-20+10}{\sqrt{3^2+(-4)^2}}|$

$=\displaystyle|\frac{-10}{\sqrt{9+16}}|$

$=\displaystyle|\frac{-10}{\sqrt{25}}|$

$=\displaystyle|\frac{-10}{5}|$

$=\displaystyle|-2|$

$=2\;\text{units}$

$\implies\;x=2$

$\implies\;x^2=4$

$\textbf{Answer:}$

$\textbf{The value of \bf\;x^2 is 4}$