Question

this question has two parts. use the information to answer.
in the circle shown, (h,k) is the center and (x,y) is on the circle.
part a
choose the two expressions that represent the lengths of the legs of the triangle.
a. h b. k c. x d. y e. x - h f. y - k
part b
which equation describes this circle?
a. h² + k² = r² b. x² + y² = r² c. (x - h)² + (y - k)² = r² d. (x + h)² + (y + k)² = r²

Explanation:

Step1: Find leg - lengths in right - triangle

In a right - triangle formed with center $(h,k)$ and a point $(x,y)$ on the circle, the horizontal leg length is the difference in $x$ - coordinates $x - h$ and the vertical leg length is the difference in $y$ - coordinates $y - k$.

Step2: Derive circle equation

The distance between the center $(h,k)$ and a point $(x,y)$ on the circle is the radius $r$. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, where $(x_1,y_1)=(h,k)$ and $(x_2,y_2)=(x,y)$, and since $d = r$, we have $r=\sqrt{(x - h)^2+(y - k)^2}$. Squaring both sides gives $(x - h)^2+(y - k)^2=r^2$.

Answer:

Part A: E. $x - h$, F. $y - k$
Part B: C. $(x - h)^2+(y - k)^2=r^2$