Circles

Explore center-radius form, completing the square, coordinate distance.

Lesson 1 of 43 parts Practice Flashcards

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Center-Radius Form

Objective: Explain center-radius form and use it to solve or interpret a problem.

A circle with center (h,k) and radius r>0 satisfies (x−h)²+(y−k)²=r². The right side is squared radius, not radius itself. Radius is a nonnegative distance.

Worked examples

Try itCircles — practice 1.1: For (x−4)²+(y−5)²=25, find the radius.Show answer

Step 1: Compare with (x−h)²+(y−k)²=r². Step 2: Here r²=25. Step 3: Take the positive root: r=5.

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Center-Radius Form

Objective: Explain center-radius form and use it to solve or interpret a problem.

A circle with center (h,k) and radius r>0 satisfies (x−h)²+(y−k)²=r². The right side is squared radius, not radius itself. Radius is a nonnegative distance.

Worked examples

Try itCircles — practice 1.1: For (x−4)²+(y−5)²=25, find the radius.Show answer

Step 1: Compare with (x−h)²+(y−k)²=r². Step 2: Here r²=25. Step 3: Take the positive root: r=5.

Completing the Square

Objective: Explain completing the square and use it to solve or interpret a problem.

Group x and y terms, complete each square, and move constants to expose the center-radius form; keep both equation sides balanced. Adding to only one side changes the circle.

Worked examples

Try itCircles — practice 2.1: For (x−2)²+(y−6)²=64, find the center x-coordinate.Show answer

Step 1: The circle form is (x−h)²+(y−k)²=r². Step 2: Compare x−2 with x−h. Step 3: The x-coordinate is h=2.

Coordinate Distance

Objective: Explain coordinate distance and use it to solve or interpret a problem.

The distance formula comes from the Pythagorean theorem; points on a circle are all r units from center. Interpret coordinate units consistently. No, a distance cannot be negative.

Worked examples

Try itCircles — practice 3.1: Find the distance from (3,7) to (6,11).Show answer

Step 1: The horizontal change is 3; vertical change is 4. Step 2: Apply √(3²+4²)=√25. Step 3: The distance is 5.

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