Question

without plotting the points: calculate the distance between each of following pair of points. find the coordinates of the midpont of line segment joining the two points (1,2) (7,10)​

Answer 1

Answers:-

Distance between the points 8 √2

Mid point co-ordinates are (4,6)

Given to find the distance between the points and mid point of line segment :-

(1 , 2 ) and (7 , 10)

SOLUTION:-

If ${(x_1 , y_1)}$ and ${(x_2, y_2)}$ are the points then

Distance between the points are :-

$\red{\boxed{\sqrt{(x_1 - x_2) {}^{2} + (y_1 - y_2) {}^{2}}}}$

The mid point were :-

$\purple{\boxed{\bigg( \dfrac{x_1 + x_2}{2} ,\dfrac{y_1 + y_2}{2} \bigg)}}$

So, According to the question,

$x_1 = 1 \\ x_2 = 7 \\ y_1 = 2 \\ y_2 = 10$

First lets find distance between the points ,

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

$= \sqrt{(1 -7 ) {}^{2} + (2 - 10) {}^{2} }$

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

$= \sqrt{64 + 64}$

$= \sqrt{128}$

$= 8 \sqrt{2}$

So, the distance between the given points is 8 √2

Now finding the midpoint co-ordinates

$\bigg( \dfrac{x_1 + x_2}{2} , \dfrac{y_1 + y_2}{2} \bigg)$

$= \bigg( \dfrac{1 + 7}{2} , \dfrac{2 + 10}{2} \bigg)$

$= \bigg( \dfrac{8}{2} , \dfrac{12}{2} \bigg)$

$= (4 , 6)$

So, the mid point co-ordinates are (4,6)

Know more formulae:-

Centroid formula:-

$\bigg(\dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3}\bigg)$

Section formula Internal division:-

$\bigg(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\bigg)$

Section formula External division:-

$\bigg(\dfrac{mx_2-nx_1}{m-n},\dfrac{my_2-ny_1}{m-n}\bigg)$