Question

use the pythagorean theorem to derive the equation of the circle centered at (5, 4) with a diameter of 10.
to which triangle should you apply the pythagorean theorem?
what is the equation of the circle?
$(x + 5)^2 + (y - 4)^2 = 25$
$(x - 5)^2 + (y - 4)^2 = 25$
$(x + 5)^2 + (y + 4)^2 = 50$
$(x - 5)^2 + (y + 4)^2 = 50$

Explanation:

Step1: Identify valid triangle

The left diagram's triangle has vertices at the circle's center $(5,4)$, the point $(x,4)$ (horizontal projection of $(x,y)$), and the general point $(x,y)$ on the circle. This forms a right triangle where the hypotenuse is the radius of the circle, so this is the triangle to use.

Step2: Calculate the radius

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

Step3: Apply Pythagorean theorem

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

Step4: Simplify the equation

$$(x - 5)^2 + (y - 4)^2 = 25$$

Answer:

1. The triangle in the left diagram (with vertices $(x,y)$, $(x,4)$, $(5,4)$)
2. $\boldsymbol{(x - 5)^2 + (y - 4)^2 = 25}$