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

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

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Answer 1 · 2017-03-01T06:35:40

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

Explanation:

If we have two points $A (x_{1}, y_{1})$ $A(x_1,y_1)$ and $B (x_{2}, y_{2})$ $B(x_2,y_2)$ , length of $¯ ¯¯¯¯ ¯ A B$ $bar(AB)$ i.e. distance between them is given by

$\sqrt{{(x_{2} - x_{1})}_{2}^{2} + {(x_{2} - x_{1})}_{2}^{2}}$ $sqrt((x_2-x_1)^2+(x_2-x_1)^2)$

and coordinates of the point $P$ $P$ that divides the segment $¯ ¯¯¯¯ ¯ A B$ $bar(AB)$ joining these two points in the ratio $l : m$ $l:m$ are

$(\frac{l x_{2} + m x_{1}}{l + m}, \frac{l x_{2} + m x_{1}}{l + m})$ $((lx_2+mx_1)/(l+m),(lx_2+mx_1)/(l+m))$

and as midpoint divided segment in ratio $1 : 1$ $1:1$ , its coordinated would be $(\frac{x_{2} + x_{1}}{2}, \frac{x_{2} + x_{1}}{2})$ $((x_2+x_1)/2,(x_2+x_1)/2)$

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

(1) the length of the segment $¯ ¯¯¯¯ ¯ A B$ $bar(AB)$ is

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

= $\sqrt{8^{2} + {(- 15)}^{2}} = \sqrt{65 + 225} = \sqrt{289} = 17$ $sqrt(8^2+(-15)^2)=sqrt(65+225)=sqrt289=17$

(2) Midpoint of $¯ ¯¯¯¯ ¯ A B$ $bar(AB)$ is $(\frac{5 - 3}{2}, \frac{- 10 - 5}{2})$ $((5-3)/2,(-10-5)/2)$ or $(1, - 7 \frac{1}{2})$ $(1,-7 1/2)$

(3) The coordinates of the point $Q$ $Q$ which splits $¯ ¯¯¯¯ ¯ A B$ $bar(AB)$ in the ratio $2 : 5$ $2:5$ are

$(\frac{2 \times 5 + 5 \times (- 3)}{7}, \frac{2 \times (- 10) + 5 \times 5}{7})$ $((2xx5+5xx(-3))/7,(2xx(-10)+5xx5)/7)$ or $(\frac{10 - 15}{7}, \frac{- 20 + 25}{7})$ $((10-15)/7,(-20+25)/7)$

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

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

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