In the xy -plane, a circle has center C with coordinates h , k . Points A and B lie

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 &#8736; A C B is a right angle. What is the length of A B &#175; ?

Accepted Answer

Choice A is correct. It's given that points A and B lie on the circle with center C . Therefore, A C &#175; and B C &#175; are both radii of the circle. Since all radii of a circle are congruent, A C &#175; is congruent to B C &#175; . The length of A C &#175; , or the distance from point A to point C , can be found using the distance formula, which gives the distance between two points, x 1 , y 1 and x 2 , y 2 , as x 1 - x 2 2 + y 1 - y 2 2 . Substituting the given coordinates of point A , h + 1 , k + 102 , for x 1 , y 1 and the given coordinates of point C , h , k , for x 2 , y 2 in the distance formula yields h + 1 - h 2 + k + 102 - k 2 , or 1 2 + 102 2 , which is equivalent to 1 + 102 , or 103 . Therefore, the length of A C &#175; is 103 and the length of B C &#175; is 103 . It's given that angle A C B is a right angle. Therefore, triangle A C B is a right triangle with legs A C &#175; and B C &#175; and hypotenuse A B &#175; . 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 103 for a and b in this equation yields 103 2 + 103 2 = c 2 , or 103 + 103 = c 2 , which is equivalent to 206 = c 2 . Taking the positive square root of both sides of this equation yields 206 = c . Therefore, the length of A B &#175; is 206 . Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This would be the length of A B &#175; if the length of A C &#175; were 103 , not 103 . Choice D is incorrect and may result from conceptual or calculation errors.

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