Question

Find the coordinates of the point which divides the line segment joining the points (-6 5) and (2 , 7) in the ratio 3:1 internally.​

Answer 1

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

Let assume that C (x, y) divides the line segment joining the points (-6, 5) and (2, 7) in the ratio 3 : 1 internally.

We know,

Section formula

Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane and R(x, y) be the point which divides PQ internally in the ratio m₁ : m₂. Then, the coordinates of R will be:

$\red{\sf\implies\boxed{\sf{ (x,y) = \bigg(\dfrac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}}, \dfrac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\bigg)}}}$

So, here

$\purple{\rm :\longmapsto\:x_1 = - 6}$

$\purple{\rm :\longmapsto\:x_2 = 2}$

$\purple{\rm :\longmapsto\:y_1 = 5}$

$\purple{\rm :\longmapsto\:y_2 = 7}$

$\purple{\rm :\longmapsto\:m_1 = 1}$

$\purple{\rm :\longmapsto\:m_2 = 3}$

So, on substituting the values in above formula, we get

$\rm :\longmapsto\:(x,y) = \bigg(\dfrac{6 - 6}{3 + 1} ,\dfrac{21 - 5}{3 + 1} \bigg)$

$\rm :\longmapsto\:(x,y) = \bigg(\dfrac{0}{4} ,\dfrac{16}{4} \bigg)$

$\red{\bf\implies \:\boxed{\sf{ \: (x,y) = (0,4) \: }}}$

Hence,

• C (0, 4) divides the line segment joining the points (-6, 5) and (2, 7) in the ratio 3 : 1 internally.

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MORE TO KNOW

Mid-point formula

Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane and R(x, y) be the mid-point of PQ. Then, the coordinates of R will be:

$\sf\implies\boxed{\tt{ R = \bigg(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\bigg)}}$

Centroid of a triangle

Centroid of a triangle is the point where the medians of the triangle meet.

Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle. Let R(x, y) be the centroid of the triangle. Then, the coordinates of R will be:

$\sf\implies\boxed{\tt{ R = \bigg(\dfrac{x_{1}+x_{2}+x_{3}}{3}, \dfrac{y_{1}+y_{2}+y_{3}}{3}\bigg)}}$