Pythagorean theorem and the equation of a circle

The  exercise appears under the Geometry Math Mission. This exercise develops the equation of a circle via the Pythagorean Theorem.

Types of Problems
There are two types of problems in this exercise:


 * 1) Find the lengths and write the equation: This problem provides a circle with a specific (though sometimes abstract) center and a numerical radius. The student is expected to find the lengths of two legs of a right triangle and use the Pythagorean Theorem to discover the equation of a circle.Ptateoac1.png
 * 2) Find the lengths inside the circle: This problem provides a circle and several right triangles with missing information inside the circle. The student is expected to correctly determine the missing lengths and supply them in the text box.Ptateoac2.png

Strategies
Knowledge of the equation of a circle can increase accuracy and efficiency, but literally the Pythagorean Theorem is all that is required to complete this exercise.
 * 1) In general, a circle with radius r and center (h,k) has equation $$ (x-h)^2+(y-k)^2=r^2 $$.
 * 2) In the drop-down menus of Find the lengths and write the equation the distances are required for the side lengths, which is why absolutely value symbols are used.

Real-life Applications

 * 1) The formula for the unit circle $$ x^2+y^2=1 $$ is fundamental in trigonometry.
 * 2) Circles can be used to model many objects in two-dimensions (carousels, balls, planets) so the equation has a plethora of applications.
 * 3) Circles and Pythagorean Theorem are common on standardized tests, such as the ACT and SAT.