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Geometry and Trigonometry Difficulty: Hard

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 $(h + 1, k + \sqrt{102})$ , and $∠ A C B$ is a right angle. What is the length of $\bar{A B}$ ?

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Explanation

Choice A is correct. It's given that points $A$ and $B$ lie on the circle with center $C$ . Therefore, $\bar{A C}$ and $\bar{B C}$ are both radii of the circle. Since all radii of a circle are congruent, $\bar{A C}$ is congruent to $\bar{B C}$ . The length of $\bar{A C}$ , or the distance from point $A$ to point $C$ , can be found using the distance formula, which gives the distance between two points, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , as $\sqrt{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}$ . Substituting the given coordinates of point $A$ , $(h + 1, k + \sqrt{102})$ , for $left parenthesis x 1 comma y 1 right parenthesis$ and the given coordinates of point $C$ , $(h, k)$ , for $left parenthesis x 2 comma y 2 right parenthesis$ in the distance formula yields $\sqrt{{(h + 1 - h)}^{2} + {(k + \sqrt{102} - k)}^{2}}$ , or $StartRoot 1 squared plus left parenthesis StartRoot 102 EndRoot right parenthesis squared EndRoot$ , which is equivalent to $StartRoot 1 plus 102 EndRoot$ , or $StartRoot 103 EndRoot$ . Therefore, the length of $\bar{A C}$ is $StartRoot 103 EndRoot$ and the length of $\bar{B C}$ is $StartRoot 103 EndRoot$ . It's given that angle $A C B$ is a right angle. Therefore, triangle $A C B$ is a right triangle with legs $\bar{A C}$ and $\bar{B C}$ and hypotenuse $\bar{A B}$ . By the Pythagorean theorem, if a right triangle has a hypotenuse with length $c$ and legs with lengths $a$ and $b$ , then $a^{2} + b^{2} = c^{2}$ . Substituting $StartRoot 103 EndRoot$ for $a$ and $b$ in this equation yields ${(\sqrt{103})}^{2} + {(\sqrt{103})}^{2} = c^{2}$ , or $103 + 103 = c^{2}$ , which is equivalent to $206 equals c squared$ . Taking the positive square root of both sides of this equation yields $StartRoot 206 EndRoot equals c$ . Therefore, the length of $\bar{A B}$ is $StartRoot 206 EndRoot$ .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This would be the length of $\bar{A B}$ if the length of $\bar{A C}$ were $103$ , not $StartRoot 103 EndRoot$ .

Choice D is incorrect and may result from conceptual or calculation errors.