Exploring the Equation of a Circle Using the Pythagorean Theorem

This lesson explores how the Pythagorean theorem can be used to graph circles by understanding the definition of a circle. It introduces the unit circle, a circle with a radius of one unit centered at the origin, and explains how right triangles can be formed within it. The lesson covers how to calculate sine and cosine values using the unit circle and addresses common mistakes students make. Finally, it generalizes the circle equation using the Pythagorean theorem, showing how x squared plus y squared equals r squared represents a circle.

What is the definition of a circle?

• A line segment with two endpoints

• A polygon with three sides

• A set of points equidistant from a center point

• A shape with four equal sides

In the unit circle, what does the x-coordinate of a point represent?

• The sine of the reference angle

• The hypotenuse of the triangle

• The cosine of the reference angle

• The tangent of the reference angle

How can you remember the order of trigonometric values in the unit circle?

• Tangent comes before sine

• Cosine comes before sine

• Sine and cosine are interchangeable

• Sine comes before cosine

What is the equation of a circle with radius r?

• x^2 + y^2 = 2r

• x^2 - y^2 = r^2

• x^2 + y^2 = r^2

• x^2 + y^2 = 1

How does the Pythagorean theorem relate to the equation of a circle?

• It determines the area of the circle

• It helps represent all points on the circle

• It is used to calculate the circumference

• It defines the diameter of the circle