# Let A be (−3,5) and B be (5,−10)). Find: (1) the length of segment bar(AB) (2) the midpoint P of bar(AB) (3) the point Q which splits bar(AB) in the ratio 2:5?

Mar 1, 2017

(1) the length of the segment $\overline{A B}$ is $17$
(2) Midpoint of $\overline{A B}$ is $\left(1 , - 7 \frac{1}{2}\right)$
(3) The coordinates of the point $Q$ which splits $\overline{A B}$ in the ratio $2 : 5$ are $\left(- \frac{5}{7} , \frac{5}{7}\right)$

#### Explanation:

If we have two points $A \left({x}_{1} , {y}_{1}\right)$ and $B \left({x}_{2} , {y}_{2}\right)$, length of $\overline{A B}$ i.e. distance between them is given by

$\sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({x}_{2} - {x}_{1}\right)}^{2}}$

and coordinates of the point $P$ that divides the segment $\overline{A B}$ joining these two points in the ratio $l : m$ are

$\left(\frac{l {x}_{2} + m {x}_{1}}{l + m} , \frac{l {x}_{2} + m {x}_{1}}{l + m}\right)$

and as midpoint divided segment in ratio $1 : 1$, its coordinated would be $\left(\frac{{x}_{2} + {x}_{1}}{2} , \frac{{x}_{2} + {x}_{1}}{2}\right)$

As we have $A \left(- 3 , 5\right)$ and $B \left(5 , - 10\right)$

(1) the length of the segment $\overline{A B}$ is

$\sqrt{{\left(5 - \left(- 3\right)\right)}^{2} + {\left(\left(- 10\right) - 5\right)}^{2}}$

= $\sqrt{{8}^{2} + {\left(- 15\right)}^{2}} = \sqrt{65 + 225} = \sqrt{289} = 17$

(2) Midpoint of $\overline{A B}$ is $\left(\frac{5 - 3}{2} , \frac{- 10 - 5}{2}\right)$ or $\left(1 , - 7 \frac{1}{2}\right)$

(3) The coordinates of the point $Q$ which splits $\overline{A B}$ in the ratio $2 : 5$ are

$\left(\frac{2 \times 5 + 5 \times \left(- 3\right)}{7} , \frac{2 \times \left(- 10\right) + 5 \times 5}{7}\right)$ or $\left(\frac{10 - 15}{7} , \frac{- 20 + 25}{7}\right)$

i.e. $\left(- \frac{5}{7} , \frac{5}{7}\right)$