Question

The distance between the points G(m+n, m-n) and H(m-n, m+n) is:
(A). $$2\sqrt{2m^{2}} $$ units
(B). $$2\sqrt{2n^{2}} $$ units
(C). $$2\sqrt{2} \left( m+n\right)$$ units
(D). $$2\sqrt{2} \left( m-n\right)$$ units

Answer 1

Hint: In this question it is given that we have to find the distance between the points G(m+n, m-n) and H(m-n, m+n). So to find the the distance between any two points we need to know about the distance formula which states that, the distance between the points $$\left( x_{1},y_{1}\right) $$ to $$\left( x_{2},y_{2}\right) $$ is $$\mathrm{d} =\sqrt{\left( x_{1}-x_{2}\right)^{2} +\left( y_{1}-y_{2}\right)^{2} }$$ ............equation(1)

Complete step-by-step answer:
Here the given points are G(m+n, m-n) and H(m-n, m+n), so by comparing the given points with $$\left( x_{1},y_{1}\right) $$ and $$\left( x_{2},y_{2}\right) $$, we can write,
$$x_{1}=m+n$$, $$y_{1}=m-n$$ and $$x_{2}=m-n$$, $$y_{2}=m+n$$
Therefore, by (1), the distance between the points G(m+n, m-n) and H(m-n, m+n),
$$\mathrm{d} =\sqrt{\left( x_{1}-x_{2}\right)^{2} +\left( y_{1}-y_{2}\right)^{2} }$$
  $$=\sqrt{\left\{ \left( m+n\right) -\left( m-n\right) \right\}^{2} +\left\{ \left( m-n\right) -\left( m+n\right) \right\}^{2} }$$
  $$=\sqrt{\left\{ m+n-m+n\right\}^{2} +\left\{ m-n-m-n\right\}^{2} }$$
  $$=\sqrt{\left( 2n\right)^{2} +\left( -2n\right)^{2} }$$
  $$=\sqrt{2^{2}\times n^{2}+\left( 2n\right)^{2} }$$ [$$\because \left( ab\right)^{n} =a^{n}\times b^{n}$$]
  $$=\sqrt{2^{2}\times n^{2}+2^{2}\times n^{2}}$$
  $$=\sqrt{4n^{2}+4n^{2}}$$
  $$=\sqrt{8n^{2}}$$
  $$=\sqrt{2\times 2\times 2\times n^{2}}$$
  $$=2\sqrt{2n^{2}}$$ [$$\because \sqrt{a\times a} =a$$]
Therefore, the distance is $$2\sqrt{2n^{2}}$$ units.
Hence the correct option is option B.

Note: While finding the distance (or the shortest distance) between two points, you have ro remember that it does not terribly matter which point is which, as long as you keep the labels (1 and 2) consistent throughout the problem.
Here $$x_{1}$$ is the horizontal coordinate (along the x axis) of Point 1, and $$x_{2}$$ is the horizontal coordinate of Point 2. $$y_{1}$$ is the vertical coordinate (along the y axis) of Point 1, and $$y_{2}$$ is the vertical coordinate of Point 2 and the above formula (1) finds the length of a line that stretches between two points: Point 1 and Point 2. The linear distance is the square root of the square of the horizontal distance plus the square of the vertical distance between two points. where horizontal distance is $$\left( x_{1}-x_{2}\right) $$ and vertical distance $$\left( y_{1}-y_{2}\right) $$.