Question

in the xy - plane, a circle has center c with coordinates (h,k). points a and b lie on the circle. point a has coordinates (h + 1,k + √102), and ∠acb is a right angle. what is the length of ab?
a √206
b 2√102
c 103√2
d 103√3

Explanation:

Step1: Calculate the length of AC

Use the distance - formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. The coordinates of $C(h,k)$ and $A(h + 1,k+\sqrt{102})$. Then $AC=\sqrt{(h + 1 - h)^2+(k+\sqrt{102}-k)^2}=\sqrt{1 + 102}=\sqrt{103}$.

Step2: Use the property of a right - angled isosceles triangle in a circle

Since $\angle ACB = 90^{\circ}$ and $AC = BC$ (radii of the same circle), by the Pythagorean theorem $AB^{2}=AC^{2}+BC^{2}$. And because $AC = BC=\sqrt{103}$, we have $AB^{2}=103+103 = 206$. Then $AB=\sqrt{206}$.

Answer:

A. $\sqrt{206}$