Question

write the coordinates of points the joint of R(-5,3) and S(19,-1) internally in the ratio 1:3.​

Answer 1

Given :-

• A line segment joining the points R(-5,3) and S(19,-1) divided by P internally in the ratio 1:3.

To Find :-

• The coordinates of point P.

Concept Used :-

Section Formula

Let us assume a line segment joining the points A and B and Let P (x, y) be any point on the line segment AB which divides AB in the ratio m : n internally, then coordinates of P is given by

$\bf \:( x, y) = \bigg(\dfrac{mx_2 + nx_1}{m + n} , \dfrac{my_2 + ny_1}{m + n} \bigg)$

$\large\underline{\bold{Solution-}}$

Given that

• A line segment joining the points R(-5,3) and S(19,-1) divided by P internally in the ratio 1 : 3.

So,

• Let coordinates of point P be (x, y) which divides the line segment RS in the ratio 1 : 3 internally.

So by Section Formula we have

$\sf \:( x, y) = \bigg(\dfrac{mx_2 + nx_1}{m + n} , \dfrac{my_2 + ny_1}{m + n} \bigg)$

Here,

• x₁ = - 5

• x₂ = 19

• y₁ = 3

• y₂ = - 1

• m = 1

• n = 3

So, on substituting all these values, we get

$\sf \: (x, y) = \bigg( \dfrac{ 19 \times 1 - 5 \times 3 }{1 + 3} , \dfrac{ - 1 \times 1 + 3 \times 3}{1 + 3} \bigg)$

$\sf \: (x, y) = \bigg( \dfrac{19 - 15}{4} , \dfrac{ - 1 + 9}{4} \bigg)$

$\sf \: (x, y) = \bigg( \dfrac{4}{4} , \dfrac{8}{4} \bigg)$

$\therefore \: \bf \: (x, \: y) \: = \: (1, \: 2)$

$\bf \: Hence, coordinates \: of \: point \: is \: (1, 2)$

Additional Information :-

Distance Formula :- is used to find the distance between two given Points.

$\underline{\boxed{\sf{\quad Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \quad}}}$

Midpoint Formula :- is used to find the midpoint of line segment.

${\underline{\boxed{\sf{\quad \bigg(\dfrac{x_1 + x_2}{2} \; , \; \dfrac{y_1 + y_2}{2} \bigg) \quad}}}}$