Question

Find the coordinates of the point which divides the line segment joining points A (-3,4)

and B (13,12) in the ratio 5:3​

Answer 1

Answer :-

The required co-ordinates is (7,9).

Step-by-step explanation:

We know,

$\bf{ \underline{Co-ordinate \: of \: P}} \\ \bigg \{ \sf{\dfrac{{m}_{1}{x}_{2} + {m}_{2}{x}_{1}}{{m}_{1} + {m}_{2}}} \: , \: \sf{\dfrac{{m}_{1}{y}_{2} + {m}_{2}{y}_{1}}{{m}_{1} + {m}_{2}}} \bigg \}$

⠀⠀⠀⠀⠀ ____________________

Let the co-ordinate of the point be P(x,y).

$\sf{\therefore \: \: \: x = \sf{\dfrac{{m}_{1}{x}_{2} + {m}_{2}{x}_{1}}{{m}_{1} + {m}_{2}}}}$

$\sf{= \sf{\dfrac{{m}_{1}{x}_{2} + {m}_{2}{x}_{1}}{{m}_{1} + {m}_{2}}}}$

$\sf{= \sf{\dfrac{5 \times 13 + 3 \times - 3}{5 + 3}}}$

$\sf{= \sf{\dfrac{65 + ( - 9)}{5 + 3}}}$

$\sf{= \sf{\dfrac{65 - 9}{5 + 3}}}$

$\sf{= \sf{\dfrac{56}{5 + 3}}}$

$\sf{= \sf{\dfrac{56}{8}}}$

$\sf{= \red{7}}$

and, $\sf{y = \sf{\dfrac{{m}_{1}{y}_{2} + {m}_{2}{y}_{1}}{{m}_{1} + {m}_{2}}}}$

$\sf{ = \dfrac{{m}_{1}{y}_{2} + {m}_{2}{y}_{1}}{{m}_{1} + {m}_{2}}}$

$\sf{ = \dfrac{5 \times 12 + 3 \times 4}{5 + 3}}$

$\sf{ = \dfrac{60 + 12}{5 + 3}}$

$\sf{ = \dfrac{72}{5 + 3}}$

$\sf{ = \dfrac{72}{8}}$

$\sf{ = \red{9}}$

Therefore, $\underline{\boxed{\textsf{\textbf{P = \{ 7,9 \}}}}}$

Answer 2

Solution:-

• Here,The coordinates of the point which divides the line segment joining A (- 3,4) and B (13,12) in the ratio 5:3.Let the us apply the section formulae of internally where m + n ≠ 0.

$\\ \large\underline{\bigstar \: \bf{Formula \: \bigstar}} \\ \\ \sf{\overline{AB} = \bigg(\frac{mx_2+nx_1}{m + n} \frac{,my_2+ny_1}{,m + n} \bigg)}$

Let A(x1,y1) = (-3,4) and B(x2,y2) = (13,12).

Given ratio m : n = 5:3

Putting the values on formula,

$\\ \\ \dashrightarrow \sf{ \bigg( \frac{(5 \times 13) + (3 \times ( - 3))}{5 + 3}, \frac{(5 \times 12) +(3 \times 4) }{5 + 3} \bigg)} \\ \\ \dashrightarrow \sf{ \bigg( \frac{65 + ( - 9)}{8} , \frac{60 + 12}{8} \bigg)} \\ \\ \dashrightarrow \sf{ \bigg( \frac{56}{8}, \frac{72}{8} \bigg)} \\ \\ \dashrightarrow \sf{7,9}$

Hence,

• The coordinate points are (7,9).

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