How to find the equation of the perpendicular bisector, the circumcenter...(Full question in the Description)

Question

Triangle DEF has vertices D(5,7), E(6,6), F(2,-2).

 

Could someone please tell and me how to find the equation of the perpendicular bisector?

 

 

 

After that I need to find the circumcenter of the triangle using the equation.

Then, I need to find the distance between the circumcenter and each vertex.

Answer 1

Hi Binu,

 

Things to remember to help solve this problem:

--Slopes are found using the formula (y₂-y₁)/(x₂-x₁)

--Perpendicular lines have slopes that are negative reciprocals of each other. If you multiply the slopes together, you get -1. For example, slopes -2 and 1/2 would be perpendicular.

--Midpoints of a segment are the average of the two endpoints

--Distance is measured with the formula d = √[(x₂-x₁)² + (y₂-y₁)²]

 

Starting with side DE, it has endpoints (5,7) and (6,6).

The midpoint of DE is ( [5+6]/2, [7+6]/2 ) = (11/2, 13/2)

The slope of DE is (6-7)/(6-5) = -1/1 = -1

The slope perpendicular to DE is 1.

The perpendicular bisector to DE has slope 1, and passes through the point (5.5, 6.5)

y-13/2 = 1(x-11/2)

y = x + 1

 

Now for side DF, with endpoints (5,7) and (2,-2)

The midpoint is ( [5+2]/2, [7+-2]/2 ) = (7/2, 5/2)

DF has slope (-2-7)/(2-5) = -9/-3 = 3

Perpendicular to DF is slope -1/3.

The perpendicular bisector to DF is

y - 5/2 = -1/3(x-7/2)

y = -1/3 x + 11/3

 

If you need/want to, you can repeat this for the last side, EF.

 

The circumcenter of the triangle is where the perpendicular bisectors intersect.

y = x + 1 = y = -1/3 x + 11/3

x + 1 = -1/3 x + 11/3       --add 1/3 x and subtract 1 from both sides

4/3 x = 8/3    -- multiply by 3/4 and reduce

x = 2

y = 3

(2, 3) is the circumcenter, label it C

 

Now you can find the distances

C(2, 3) to D(5, 7)

d = √[(5-2)² + (7-3)²]

d = √[3² + 4²]

d = √[25] = 5.

 

Repeat for distances CE and CF. They should all be the same.