Question

use the pythagorean theorem to derive the equation of the circle centered at (6, 4) with a diameter of 10.
to which triangle should you apply the pythagorean theorem?
what is the equation of the circle?
$(x - 6)^2 + (y - 4)^2 = 25$
$(x + 6)^2 + (y + 4)^2 = 25$
$(x - 6)^2 + (y - 4)^2 = 50$
$(x + 6)^2 + (y + 4)^2 = 50$

Explanation:

Step1: Identify correct triangle

The right triangle with vertices $(6,4)$, $(x,4)$, and $(x,y)$ (the right panel triangle) is the correct one, as it uses the center, a point on the circle, and a horizontal/vertical leg to form a right triangle where the radius is the hypotenuse.

Step2: Calculate the radius

The diameter is 10, so radius $r = \frac{10}{2} = 5$.

Step3: Apply Pythagorean theorem

The horizontal leg length is $|x - 6|$, vertical leg length is $|y - 4|$, hypotenuse is $r=5$. By Pythagoras:
$$(x - 6)^2 + (y - 4)^2 = 5^2$$

Step4: Simplify the equation

$$(x - 6)^2 + (y - 4)^2 = 25$$

Answer:

1. The correct triangle is the one in the right panel (vertices $(6,4)$, $(x,4)$, $(x,y)$)
2. $\boldsymbol{(x - 6)^2 + (y - 4)^2 = 25}$