Diameter endpoints call for the midpoint: it is the circle’s center. Use Desmos’s midpoint command to find it, then use the distance from the center to either endpoint as the radius. Put the center and radius into standard form. The tempting slip is using the full diameter as the radius, which makes the circle too large.
Hints
- Hint 1
A diameter passes through the center, so its midpoint is the center. You can average the endpoints’ coordinates separately, or use Desmos’s midpoint command. What point is halfway between them?
- Hint 2
Connect the center to either endpoint. That segment is a radius. Find its horizontal and vertical changes. Do they make a familiar right triangle?
- Hint 3
A circle’s standard form is , where is the center. The equation needs the radius squared, and the signs inside the parentheses are opposite the center’s coordinates.
Step-by-step
Find the center, then the radius
Step 1Find the center of the diameter
A diameter passes through the circle’s center, so the center is halfway between its endpoints. Type , , and in Desmos. Click the midpoint it plots; its coordinates show . So the center is .
- Step 2
Measure the changes from the center
Go from the center to endpoint . The horizontal change is , and the vertical change is . These perpendicular moves are the two legs of a right triangle; the segment back to the center is the radius.
- Step 3
Recognize the radius
A -- right triangle has a slanted side of length , so . Don’t use the distance all the way from to as the radius: that is the diameter, twice as long.
- Step 4
Write the circle’s equation
In standard form, , the center is . Substitute the center and radius :
Square the radius on the right:
That is the equation of the circle. Choice D.
Lessons that teach this
- SAT Geometry and TrigonometryIntermediateCoreBuild circles from coordinate information
- SAT Geometry and TrigonometryBeginnerCoreRead and write circle equations
- SAT Geometry and TrigonometryBeginnerApply the Pythagorean theorem
- DesmosAdvancedCoreCircle geometry with distance and midpoint
- DesmosIntermediateCircle equations in the graph