We can work with **circles** in the coordinate plane.

The Pythagorean theorem, or distance formula, can help us find distances in the coordinate plane.

\displaystyle d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}

\bm{d}

distance between the two points

\bm{\left(x_{1},\,y_{1}\right)}

coordinates of the first point

\bm{\left(x_{2},\,y_{2}\right)}

coordinates of the second point

The center is the midpoint of the diameter so we can use the midpoint formula to find the center when we know the endpoints of the diameter.

\displaystyle \left(x_M,\,y_M\right)=\left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2} \right)

\bm{\left(x_M,\,y_M\right)}

coordinates of the midpoint

\bm{\left(x_1,\,y_1\right)}

coordinates of the first endpoint

\bm{\left(x_2,\,y_2\right)}

coordinates of the second endpoint

Given center (h,\,k), we can determine whether points in the coordinate plane lie on a circle.

- If \left(x_{1}-h\right)^{2}+\left(y_{1}-k\right)^{2}<r^{2} then \left( x_{1},\,y_{1} \right) is inside the circle
- If \left(x_{1}-h\right)^{2}+\left(y_{1}-k\right)^{2}=r^{2} then \left( x_{1},\,y_{1} \right) is on the circle
- If \left(x_{1}-h\right)^{2}+\left(y_{1}-k\right)^{2}>r^{2} then \left( x_{1},\,y_{1} \right) is outside the circle

The given circle has a diameter with endpoints (-1,\, 3.65) and (5,\, -1.65):

a

Find the center of the circle.

Worked Solution

b

Find the length of the diameter of the circle.

Worked Solution

Find the center of the circle whose endpoints of a diameter are \left(-1.5,\, 4\right) and \left(4.5,\,-2\right).

Worked Solution

Determine if the point (4,\,3) lies on the circle with a center at (1,\,1) and radius of 3.

Worked Solution

A circle has a center of (-1,\,-4) and contains the point (4,\,8)

a

Find the length of the radius.

Worked Solution

b

Find the diameter of the circle.

Worked Solution

Idea summary

We can use the distance formula and midpoint formula to help determine the length and location of points on the circle and the center, radius, and diameter of a circle.

Distance formula

\displaystyle d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}

\bm{d}

distance between the two points

\bm{\left(x_{1},y_{1}\right)}

coordinates of the first point

\bm{\left(x_{2},\,y_{2}\right)}

coordinates of the second point

Midpoint formula

\displaystyle \left(x_M,\,y_M\right)=\left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2} \right)

\bm{\left(x_M,\,y_M\right)}

coordinates of the midpoint

\bm{\left(x_1,\,y_1\right)}

coordinates of the first endpoint

\bm{\left(x_2,\,y_2\right)}

coordinates of the second endpoint