Question

If A ( 7, 10), B( -2, k) and C(3, -4) are vertices of a right angled triangle at B, then k=

Answer 1

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

Given that, In right triangle, ABC, right angled at B,

Coordinates of A is (7, 10)

Coordinates B is (- 2, k)

Coordinates of C is (3, - 4)

So, using Pythagoras Theorem, we have

$\rm \: {AB}^{2} + {BC}^{2} = {AC}^{2}$

So, using Distance formula, we get

$\rm \: {(7 + 2)}^{2} + {(10 - k)}^{2} + {( - 2 - 3)}^{2} + {(k + 4)}^{2} = {(7 - 3)}^{2} + {(10 + 4)}^{2}$

$\rm \: 81 + {(10 - k)}^{2} + 25 + {(k + 4)}^{2} = 16 + 196$

$\rm \: 106 + {(10 - k)}^{2} + {(k + 4)}^{2} = 212$

$\rm \: {(10 - k)}^{2} + {(k + 4)}^{2} = 106$

$\rm \: 100 + {k}^{2} - 20k + {k}^{2} + 16 + 8k = 106$

$\rm \: 2{k}^{2} + 116 - 12k = 106$

$\rm \: 2{k}^{2} + 116 - 12k - 106 = 0$

$\rm \: 2{k}^{2} - 12k + 10 = 0$

$\rm \: 2({k}^{2} - 6k + 5) = 0$

$\rm \:{k}^{2} - 6k + 5 = 0$

$\rm \:{k}^{2} - 5k - k + 5 = 0$

$\rm \: k(k - 5) - 1(k - 5) = 0$

$\rm \: (k - 5)(k - 1) = 0$

$\rm\implies \:k \: = \: 5 \: \: or \: \: k \: = \: 1$

$\rule{190pt}{2pt}$

Formula Used :-

Distance Formula

Let A(x₁, y₁) and B(x₂, y₂) be two points in the cartesian plane, then distance between A and B is given by

$\begin{gathered}\boxed{\tt{ AB \: = \sqrt{ {(x_{1} - x_{2}) }^{2} + {(y_{2} - y_{1})}^{2} }}} \\ \end{gathered}$

$\rule{190pt}{2pt}$

Additional Information :-

1. Section formula

Let A(x₁, y₁) and B(x₂, y₂) be two points in the cartesian plane and C(x, y) be the point which divides AB internally in the ratio m₁ : m₂, then the coordinates of C is given by

$\begin{gathered} \boxed{\tt{ (x, y) = \bigg(\dfrac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}}, \dfrac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\bigg)}} \\ \end{gathered}$

2. Mid-point formula

Let A(x₁, y₁) and B(x₂, y₂) be two points in the coordinate plane and C(x, y) be the mid-point of AB, then the coordinates of C is given by

$\begin{gathered}\boxed{\tt{ (x,y) = \bigg(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\bigg)}} \\ \end{gathered}$

3. Centroid of a triangle

Centroid of a triangle is defined as the point at which the medians of the triangle meet and is represented by the symbol G.

Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle and G(x, y) be the centroid of the triangle, then the coordinates of G is given by

$\begin{gathered}\boxed{\tt{ (x, y) = \bigg(\dfrac{x_{1}+x_{2}+x_{3}}{3}, \dfrac{y_{1}+y_{2}+y_{3}}{3}\bigg)}} \\ \end{gathered}$

4. Area of a triangle

Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle, then the area of triangle is given by

$\begin{gathered}\boxed{\tt{ Area =\dfrac{1}{2}\bigg|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\bigg|}} \\ \end{gathered}$