Question

∆ABC has vertices A(–2, 0), B(0, 8), and C(4, 2)

Find the equations of the perpendicular bisectors of the sides of the ∆ABC.

Find the coordinates of the point of concurrency of the perpendicular bisectors (P) .

Answer 1

Part 1) The equation of the perpendicular bisector side AB is
`y=-(1)/(4)x+(15)/(4)`

Part 2) The equation of the perpendicular bisector side BC is
`y=(2)/(3)x+(11)/(3)`

Part 3) The equation of the perpendicular bisector side AC is
`y=-3x+4`

Part 4) The coordinates of the point P(0.091,3.727)

Explanation:

Part 1) Find the equation of the perpendicular bisector side AB

we have

A(–2, 0), B(0, 8)

step 1

Find the slope AB

The formula to calculate the slope between two points is equal to

`m=(y2-y1)/(x2-x1)`

substitute the values

`m=(8-0)/(0+2)`

`m=4`

step 2

Find the slope of the perpendicular line to side AB

Remember that

If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is equal to -1)

therefore

The slope is equal to

`m=-(1)/(4)`

step 3

Find the midpoint AB

The formula to calculate the midpoint between two points is equal to

`M((x1+x2)/(2),(y1+y2)/(2))`

substitute the values

`M((-2+0)/(2),(0+8)/(2))`

`M(-1,4)`

step 4

Find the equation of the perpendicular bisectors of AB

the slope is
`m=-(1)/(4)`

passes through the point
`(-1,4)`

The equation in slope intercept form is equal to

`y=mx+b`

substitute

`4=(-(1)/(4))(-1)+b`

solve for b

`b=4-(1)/(4)`

`b=(15)/(4)`

so

`y=-(1)/(4)x+(15)/(4)`

Part 2) Find the equation of the perpendicular bisector side BC

we have

B(0, 8) and C(4, 2)

step 1

Find the slope BC

The formula to calculate the slope between two points is equal to

`m=(y2-y1)/(x2-x1)`

substitute the values

`m=(2-8)/(4-0)`

`m=-(3)/(2)`

step 2

Find the slope of the perpendicular line to side BC

Remember that

If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is equal to -1)

therefore

The slope is equal to

`m=(2)/(3)`

step 3

Find the midpoint BC

The formula to calculate the midpoint between two points is equal to

`M((x1+x2)/(2),(y1+y2)/(2))`

substitute the values

`M((0+4)/(2),(8+2)/(2))`

`M(2,5)`

step 4

Find the equation of the perpendicular bisectors of BC

the slope is
`m=(2)/(3)`

passes through the point
`(2,5)`

The equation in slope intercept form is equal to

`y=mx+b`

substitute

`5=((2)/(3))(2)+b`

solve for b

`b=5-(4)/(3)`

`b=(11)/(3)`

so

`y=(2)/(3)x+(11)/(3)`

Part 3) Find the equation of the perpendicular bisector side AC

we have

A(–2, 0) and C(4, 2)

step 1

Find the slope AC

The formula to calculate the slope between two points is equal to

`m=(y2-y1)/(x2-x1)`

substitute the values

`m=(2-0)/(4+2)`

`m=(1)/(3)`

step 2

Find the slope of the perpendicular line to side AC

Remember that

If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is equal to -1)

therefore

The slope is equal to

`m=-3`

step 3

Find the midpoint AC

The formula to calculate the midpoint between two points is equal to

`M((x1+x2)/(2),(y1+y2)/(2))`

substitute the values

`M((-2+4)/(2),(0+2)/(2))`

`M(1,1)`

step 4

Find the equation of the perpendicular bisectors of AC

the slope is
`m=-3`

passes through the point
`(1,1)`

The equation in slope intercept form is equal to

`y=mx+b`

substitute

`1=(-3)(1)+b`

solve for b

`b=1+3`

`b=4`

so

`y=-3x+4`

Part 4) Find the coordinates of the point of concurrency of the perpendicular bisectors (P)

we know that

The point of concurrency of the perpendicular bisectors is called the circumcenter.

Solve by graphing

using a graphing tool

the point of concurrency of the perpendicular bisectors is P(0.091,3.727)

see the attached figure