What are the equations of the perpendicular bisectors on the triangle between the three points?
Center of circle: (10,10) Radius: 10 units
Practice makes perfect
Let's plot the given points and the circular pathway.
To find the center of a circle that passes through three known points, we can draw a triangle between the points and then proceed to find the triangle's circumcenter. This works because the circumcenter is equidistant from the triangles vertices.
Finding the circumcenter
To find the circumcenter, we have to graph at least two perpendicular bisectors. Start by finding the perpendicular bisector to PT. We construct two arcs, one using T as center, and another using P as center using the compass setting is the same for both arcs and that the radius is greater than half the length of PT.
The perpendicular bisector is the segment where the two arcs intersect.
Let's repeat the procedure for TN.
Finally, we will go ahead and plot the triangle's circumcenter which is also the center of our circle.
Finding the radius
To determine the circle's radius, we need to know the coordinates of the circle's midpoint. We can find this point if we determine equations for the perpendicular bisectors and then proceed to find their point of intersection. Let's first calculate the slope of PT and NT.
Side
Points
y_2-y_1/x_2-x_1
m
PT
( 10,20), ( 2,4)
20- 4/10- 2
2
TN
( 16,2), ( 2,4)
2- 4/16- 2
- 1/7
The slopes of perpendicular lines are opposite reciprocals. Knowing the slopes of PT and NT we can write two equations that describe the perpendicular bisectors to these sides:
Perpendicular bisector PT:& y=- 1/2x+b
Perpendicular bisector TN:& y= 7x+b
To finish the equations, we need to substitute a point through which the perpendicular bisectors passes. Since the bisectors divide their respective sides in half, we can find such a point by calculating the midpoints of PT and TN.
Side
Points
M(x_1+x_2/2,y_1+y_2/2)
Midpoint
PT
( 2,4), ( 10,20)
M(2+ 10/2,4+ 20/2)
M(6,12)
TN
( 2,4), ( 16,2)
M(2+ 16/2,4+ 2/2)
M(9,3)
By substituting M( 6,12) in the equation for the perpendicular bisector to PT, and M( 9,3) in the equation for the perpendicular bisector to TN, we can solve for their y-intercepts.
12&=- 1/2( 6)+b ⇒ b=15.
3&=7( 9)+b ⇒ b=- 60.
When we know the equations of the perpendicular bisectors, we can equate them to find their point of intersection.
Finally, we will calculate the corresponding y-coordinate by substituting x= 10 in either equation.
y=7( 10)-60 ⇒ y=10.
Now we are mathematically sure that the circle's midpoint is at (10,10). Finally, we will use the Distance Formula to calculate the distance between the midpoint and an arbitrary vertex of the triangle.
