Question

Prove that the distance between the origin and the point $(-6,-8)$ is twice the distance between the points $(4,0)$ and $(0,3)$.

Answer 1

Let $d_1$ and $d_2$ be the distance between the points $(0,0)\&(-6,-8)$ and $(4,0)\&(0,3)$ respectively.

To prove:- $d_1=2d_2$

As we know that the distance between the points $(x_1,y_1)$ and $(x_2,y_2)$ is given as-

$d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Therefore,

$d_1=\sqrt{{(-6-0)}^2+{(-8-0)}^2}$

$d_1=\sqrt{{(-6)}^2+{(-8)}^2}$

$\Rightarrow d_1=\sqrt{36+64}$

$\Rightarrow d_1=\sqrt{100}=10$

$\Rightarrow d_1=2\times 5.....\left(1\right)$

Now,

$d_2=\sqrt{{(3-0)}^2+{(0-4)}^2}$

$d_2=\sqrt{{\left(3\right)}^2+{(-4)}^2}$

$\Rightarrow d_2=\sqrt{9+16}$

$\Rightarrow d_2=\sqrt{25}=5.....\left(1\right)$

From ${eq}^n\left(1\right)\&\left(2\right)$, we have

$d_1=2\times d_2(\because d_2=5)$

Hence proved.