Question

if ABC are points(-1,1),(1,3) and (3,a) respectively and if AB=BC then find a

Answer 1

Given that ABC is a triangle. Coordinate of the corner points are A(-1,1), B(1,3) and C(3,a)

Given that AB=BC

means side length of AB and BC are equal so we can use distance formula to find the distance of AB and BC and set them equal to each other

Distance between two points (x1,y1) and (x2,y2) is given by formula:

$Distance=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$

So $AB=\sqrt{\left(3-1\right)^2+\left(1--1\right)^2}$

Similarly $BC=\sqrt{\left(a-3\right)^2+\left(3-1\right)^2}$

Set both equal to each other

$\sqrt{\left(3-1\right)^2+\left(1--1\right)^2}=\sqrt{\left(a-3\right)^2+\left(3-1\right)^2}$

$\sqrt{2^2+2^2}=\sqrt{\left(a-3\right)^2+2^2}$

$\sqrt{4+4}=\sqrt{\left(a-3\right)^2+4}$

take square of both sides

$4+4=\left(a-3\right)^2+4$

$4=\left(a-3\right)^2$

$\pm 2=a-3$

+2=a-3 and -2=a-3

5=a and 1=a

Hence a=1 and a=5 are the final answer.