Understanding the Pythagorean Theorem

Coordinate Geometry Fundamentals

The coordinate plane is divided into four quadrants (I, II, III, IV) by the x-axis and y-axis, which intersect at the origin. This system gives us a way to pinpoint exact locations using ordered pairs (x,y).

The Pythagorean Theorem $a² + b² = c²$ is the foundation for the distance formula, which lets you find the distance between any two points: d = √$(x₂-x₁)² + (y₂-y₁)²$. When you need to find the midpoint between two locations, use the midpoint formula: $[(x₁+x₂)/2], [(y₁+y₂)/2]$ - just average the x-coordinates and y-coordinates.

When sketching graphs, follow these steps: isolate a variable when possible, create a table of values, plot the points, and connect them with a smooth curve or line. You can identify symmetry in three ways: x-axis symmetry $x,-y$, y-axis symmetry $-x,y$, or origin symmetry $-x,-y$.

Pro Tip: Test for symmetry algebraically by substituting -y for y (x-axis symmetry), -x for x (y-axis symmetry), or both -x and -y (origin symmetry). If you get an equivalent equation, symmetry exists!

The standard form of a circle equation is $x-h$² + $y-k$² = r², where (h,k) is the center and r is the radius. When the center is at the origin, this simplifies to x² + y² = r². This formula comes directly from applying the distance formula from the center to any point on the circle.