Question

4 KM is a straight line of 13 units. If K has the co-ordinates (2, 5) and M has the co-ordinates (x,-7), find the values of x.​

Answer 1

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

Given that,

• Coordinates of K is (2, 5)

• Coordinates of M is (x, - 7)

• Length of KM is 13 units.

We know

Distance Formula :- Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane, then distance between P and Q is given by

$\purple{\rm :\longmapsto\:\boxed{\tt{ PQ \: = \: \sqrt{ {(x_{1} - x_{2}) }^{2} + {(y_{2} - y_{1})}^{2} }}}}$

So, Here,

• • x₁ = 2

• • x₂ = x

• • y₁ = 5

• • y₂ = - 7

So, distance between these points is,

$\rm :\longmapsto\: \sqrt{ {(x - 2)}^{2} + {( - 7 - 5)}^{2} } = 13$

$\rm :\longmapsto\: \sqrt{ {(x - 2)}^{2} + {( - 12)}^{2} } = 13$

$\rm :\longmapsto\: \sqrt{ {(x - 2)}^{2} + 144 } = 13$

On squaring both sides, we get

$\rm :\longmapsto\: {(x - 2)}^{2} + 144 = 169$

$\rm :\longmapsto\: {(x - 2)}^{2} = 169 - 144$

$\rm :\longmapsto\: {(x - 2)}^{2} = 25$

$\rm :\longmapsto\: {(x - 2)}^{2} = {5}^{2}$

$\rm :\longmapsto\:x - 2 \: = \: \pm \: 5$

$\bf\implies \:x = 7 \: \: or \: \: x = - 3$

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Learn More :-

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:

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

2. 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:

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

3. 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:

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