Question

If P (a/3, 4) is the mid-point of the line segment joining the points Q(-4, 7) and R(-2, 1), then the value of ‘a’ is

Answer 1

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

Given that,

P (a/3, 4) is the mid-point of the line segment joining the points Q(-4, 7) and R(-2, 1).

We know, Midpoint 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 line segment joining the points P and Q. Then, the coordinates of R will be given as

$\rm :\longmapsto\:\boxed{ \tt{ \: (x,y) \: = \: \bigg(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\bigg) \: }}$

Here,

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:-\begin{cases} &\sf{x = \dfrac{a}{3} } \\ \\ &\sf{y = 4}\\ \\ &\sf{x_1 = - 4}\\ \\ &\sf{x_2 = - 2}\\ \\ &\sf{y_1 = 7}\\ \\ &\sf{y_2 = 1} \end{cases}\end{gathered}\end{gathered}$

So, on substituting the values, we get

$\rm :\longmapsto\:\bigg(\dfrac{a}{3} ,4 \bigg) = \bigg(\dfrac{ - 4 - 2}{2}, \: \dfrac{7 + 1}{2} \bigg)$

$\rm :\longmapsto\:\bigg(\dfrac{a}{3} ,4 \bigg) = \bigg(\dfrac{ - 6}{2}, \: \dfrac{8}{2} \bigg)$

$\rm :\longmapsto\:\bigg(\dfrac{a}{3} ,4 \bigg) = \bigg( - 3, \: 4 \bigg)$

So, on comparing x - coordinate, we get

$\rm :\longmapsto\:\dfrac{a}{3} = - 3$

$\bf\implies \:a \: = \: - \: 9$

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More to know :-

1. 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:

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

2. 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 G(x, y) be the centroid of the triangle. Then, the coordinates of G is given by :

$\boxed{ \tt{ \: G \: = \bigg(\dfrac{x_{1}+x_{2}+x_{3}}{3}, \dfrac{y_{1}+y_{2}+y_{3}}{3}\bigg) \: }}$

3. Distance Formula

Let A(x₁, y₁), B(x₂, y₂) points in the cartesian plane. Distance between two points is calculated by using the formula given below :

$\boxed{ \tt{ \: D = \sqrt{ {(x_{1} - x_{2}) }^{2} + {(y_{2} - y_{1})}^{2} } \: }}$

Answer 2

Answer:

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

Given that,

P (a/3, 4) is the mid-point of the line segment joining the points Q(-4, 7) and R(-2, 1).

We know, Midpoint 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 line segment joining the points P and Q. Then, the coordinates of R will be given as

$\rm :\longmapsto\:\boxed{ \tt{ \: (x,y) \: = \: \bigg(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\bigg) \: }}$

Here,

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:-\begin{cases} &\sf{x = \dfrac{a}{3} } \\ \\ &\sf{y = 4}\\ \\ &\sf{x_1 = - 4}\\ \\ &\sf{x_2 = - 2}\\ \\ &\sf{y_1 = 7}\\ \\ &\sf{y_2 = 1} \end{cases}\end{gathered}\end{gathered}$

So, on substituting the values, we get

$\rm :\longmapsto\:\bigg(\dfrac{a}{3} ,4 \bigg) = \bigg(\dfrac{ - 4 - 2}{2}, \: \dfrac{7 + 1}{2} \bigg)$

$\rm :\longmapsto\:\bigg(\dfrac{a}{3} ,4 \bigg) = \bigg(\dfrac{ - 6}{2}, \: \dfrac{8}{2} \bigg)$

$\rm :\longmapsto\:\bigg(\dfrac{a}{3} ,4 \bigg) = \bigg( - 3, \: 4 \bigg)$

So, on comparing x - coordinate, we get

$\rm :\longmapsto\:\dfrac{a}{3} = - 3$

$\bf\implies \:a \: = \: - \: 9$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

More to know :-

1. 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:

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

2. 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 G(x, y) be the centroid of the triangle. Then, the coordinates of G is given by :

$\boxed{ \tt{ \: G \: = \bigg(\dfrac{x_{1}+x_{2}+x_{3}}{3}, \dfrac{y_{1}+y_{2}+y_{3}}{3}\bigg) \: }}$

3. Distance Formula

Let A(x₁, y₁), B(x₂, y₂) points in the cartesian plane. Distance between two points is calculated by using the formula given below :

$\boxed{ \tt{ \: D = \sqrt{ {(x_{1} - x_{2}) }^{2} + {(y_{2} - y_{1})}^{2} } \: }}$