Question

use the pythagorean theorem to derive the equation of the circle centered at (6, 5) with a diameter of 6.
to which triangle should you apply the pythagorean theorem?
what is the equation of the circle?
$(x + 6)^2 + (y + 5)^2 = 3$ $(x - 6)^2 + (y - 5)^2 = 3$
$(x + 6)^2 + (y + 5)^2 = 9$ $(x - 6)^2 + (y - 5)^2 = 9$

Explanation:

Step1: Identify correct triangle

The right triangle connects the circle's center $(6,5)$, a point $(x,5)$ on the horizontal line from the center, and any point $(x,y)$ on the circle (second diagram). This forms a right triangle where the legs are the horizontal/vertical distances from the center to $(x,y)$, and the hypotenuse is the radius.

Step2: Calculate the radius

The diameter is 6, so radius $r = \frac{6}{2} = 3$.

Step3: Apply Pythagorean theorem

Horizontal leg length: $|x - 6|$, vertical leg length: $|y - 5|$, hypotenuse = $r=3$.
By Pythagorean theorem:
$$(x - 6)^2 + (y - 5)^2 = 3^2$$

Step4: Simplify the equation

$$(x - 6)^2 + (y - 5)^2 = 9$$

Answer:

1. Correct triangle: The triangle in the second diagram (connecting $(6,5)$, $(x,5)$, and $(x,y)$)
2. Equation of the circle: $(x - 6)^2 + (y - 5)^2 = 9$