Circles in the x-y Plane
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Coordinate Geometry
Summary
A circle centered at (h, k) with radius r has the equation (x - h)² + (y - k)² = r², which comes directly from the Pythagorean theorem. ACT success depends on understanding how the center and radius appear in the equation and then applying that relationship to coordinate-geometry problems involving tangent squares and vertex coordinates.
- A circle at the origin follows x² + y² = r², and shifting the center away from the origin changes the equation by replacing x and y with distances from the new center coordinates.
- Using a right triangle from the center (h, k) to a point (x, y) on the circle shows that the horizontal and vertical distances are x - h and y - k, so the general equation becomes (x - h)² + (y - k)² = r².
- The most important skill is not memorizing the formula mechanically but understanding why it works, since that makes it easier to identify the center, radius, and sign changes correctly.
- In a square with vertices (-1, 1), (3, 1), (3, -3), and (-1, -3), a circle tangent to all four sides has center (1, -1) and radius 2, giving the equation (x - 1)² + (y + 1)² = 4.
- From a circle equation such as (x + 4)² + (y - 2)² = 25, the center is (-4, 2) and the radius is 5; moving 5 units left, right, up, and down gives the tangent square's sides and vertices.
- To maximize the sum of x- and y-coordinates among the square's vertices, choose the upper-right vertex because it has the greatest x-value and greatest y-value; in the example, that sum is 8.
Chapters
00:00
Deriving the General Circle Equation
00:21
Using the Pythagorean Theorem with a Shifted Center
01:56
Applying the Formula to a Tangent Circle Inside a Square
03:20
Finding a Surrounding Square from a Circle Equation
06:29
Core Rule to Remember for ACT Geometry