Question

A perpendicular bisector, CD is drawn through point C on AB. 

If the coordinates of point A are (-3, 2) and the coordinates of point B are (7, 6), the x-intercept of CD is[ A(3,0) B(18/5,0) C(9,0) D(45/2,0) ] . Point [ A(-52,117) B(-20,57) C(32,-71) D(-54,-128) ] lies on CD.

Answer 1

The x-intercept of CD is B(18/5,0). The point C(32,-71) lies on the line CD.

Explanation:

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Answer 2

The x-intercept of CD is B(18/5,0). The point C(32,-71) lies on the line CD.

Explanation:

Given information: CD is perpendicular bisector of AB.

The coordinates of point A are (-3, 2) and the coordinates of point B are (7, 6).

Midpoint of AB is C.

`C=((x_1+x_2)/(2),(y_1+y_2)/(2))=((7-3)/(2),(2+6)/(2))=(2,4)`

The coordinates of C are (2,4).

The slope of line AB is

`m_1=(y_2-y_1)/(x_2-x_1)=(6-2)/(7-(-3))=(4)/(10)=(2)/(5)`

The product of slopes of two perpendicular lines is -1. Since the line CD is perpendicular to AB, therefore the slope of CD is

`m_2=-(5)/(2)`

The point slope form of a line is

`y-y_1=m(x-x_1)`

The slope of line CD is
`-(5)/(2)` and the line passing through the point (2,4), the equation of line CD is

`y-4=-(5)/(2)(x-2)`

`y=-(5)/(2)x+5+4`

`y=-(5)/(2)x+9` .... (1)

The equation of CD is
`y=-(5)/(2)x+9`.

Put y=0, to find the x-intercept.

`0=-(5)/(2)x+9`

`(5)/(2)x=9`

`x=(18)/(5)`

Therefore the x-intercept of CD is B(18/5,0).

Put x=-52 in equation (1).

`y=-(5)/(2)(-52)+9=139`

Put x=-20 in equation (1).

`y=-(5)/(2)(-20)+9=59`

Put x=32 in equation (1).

`y=-(5)/(2)(32)+9=-71`

Put x=-54 in equation (1).

`y=-(5)/(2)(-54)+9=144`

Only point (32,-71) satisfies the equation of CD. Therefore the point C(32,-71) lies on the line CD.