Question

The perpendicular bisector of a line segment AB passes through the origin . if coordinates of (4,0), then find the coordinate of B and the length of AB​

Answer 1

The coordinate of B is (-4, 0), and the length of AB = 8 units

Given:

The perpendicular bisector of a line segment AB passes through the origin and the coordinate of A (4, 0)

To find:

Find the coordinate of B and the length of AB​  

Solution:  

Formula used:

Distance between two points = [ (x₁+x₂)/2,  (y₁+y₂)/2 ]  

Length of a line segment = √(y₂-y₁)²+(x₂-x₁)²

Perpendicular Bisector:

A perpendicular Bisector is a line that cuts the line segment exactly into two parts and makes 90° with the line segment. A perpendicular Bisector of a line segment always passes through the midpoint of the line segment.

     

Given that the perpendicular bisector of Line segment AB passes through the origin (0, 0)  

Which means The midpoint of AB = (0, 0)  

Let the coordinates of B be (x, y)

=> Midpoint of A(4, 0) and B(x, y) = (0, 0)

=> $[\frac{4+x}{2} , \frac{0+y}{2} ] = (0, 0)$  

=> $\frac{4+x}{2} = 0$     and       $\frac{0+y}{2} = 0$    

=> 4 + x = 0               0 + y = 0

=> x = - 4                    y = 0

Therefore,

The coordinate of B(x, y) = (-4, 0)

Length AB = Distance between A (4, 0) and B (-4, 0)  

AB = √(0 - 0)² + (4 + 4)² = √8² = 8 units  

Therefore,

The coordinate of B is (-4, 0), and the length of AB = 8 units

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