Question

njecture about the Equation of a Circle
On a coordinate plane, circle Q is centered at the origin with radius r. Triangle P Q S is shown. Point Q is at (0, 0) and points P is at (x, y). The length of Q P is r, the length of P S is y, and the length of Q S is x. Angle P S Q is a right angle.

Circle Q is centered at the origin with radius r. Point P(x, y) lies on circle Q. Make a conjecture. How can you find an equation relating the radius to the coordinates of point P? Check all that apply.

Notice that ΔPQS forms a right triangle.
Because ΔPQS is a right triangle, apply the Pythagorean theorem.
x² + y² = r²

Answer 1

Conjecture: x^2 + y^2 = r^2.

Reason 1 (right triangle + Pythagorean theorem): PQ = r is the hypotenuse of right triangle PQS and QS = x, PS = y are the legs, so x^2 + y^2 = r^2.

Reason 2 (distance formula): The distance from the origin Q(0,0) to P(x,y) is r, so r = sqrt((x-0)^2 + (y-0)^2) = sqrt(x^2 + y^2). Squaring gives x^2 + y^2 = r^2.

This is the standard equation of a circle centered at the origin; equivalently y = ±sqrt(r^2 - x^2).