First slide

Equation of line in Straight lines

Question

The equation of base of an equilateral triangle is $4 x + 3 y - 25 = 0$ and one vertex is origin then the length of the side is

Moderate

A

$10 \sqrt{3}$
B

$\frac{10}{9}$
C

$\frac{10 \sqrt{3}}{9}$
D

$\frac{10 \sqrt{3}}{3}$

Solution

$The perpendicular distance from a point P (x_{1}, y_{1}) to the line a x + b y + c = 0 is |\frac{a x_{1} + b y_{1} + c}{\sqrt{a^{2} + b^{2}}}|$
$The perpendicular distance from a point (0, 0) to the line 4 x + 3 y - 25 = 0 is$
$|\frac{4 (0) + 3 (0) - 25}{\sqrt{16 + 9}}| = \frac{25}{5} = 5$
$This is height h of the equilateral triangle.$
$If a be the side of an equilateral triangle, then a = \frac{2 h}{\sqrt{3}}$
Hence,
$\begin{matrix} a = \frac{2 (5)}{\sqrt{3}} \\ = \frac{10 \sqrt{3}}{3} \end{matrix}$
$Therefore, the length of the side of equilateral triangle is a = \frac{10 \sqrt{3}}{3}$

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