sat-dog

Choice A is correct. It’s given that points A and B lie on the circle with center C. Therefore, AC and BC are both radii of the circle. Since all radii of a circle are congruent, AC is congruent to BC. The length of AC, or the distance from point A to point C, can be found using the distance formula, which gives the distance between two points, ^x1 , y1 h and ^x2 , y2 h, as ^x1- x2 h 2+ ^y1- y2 h 2 . Substituting the given coordinates of point A, ^h +1, k + 102 h, for ^x1 , y1 h and , the given coordinates of point C, ^h,kh, for ^x2 y2 h in the distance formula yields +1- 2+ + 102- 2 12+ 102 2 ^h hh ^k kh , or ^ h , which is equivalent to 1+102 103 103 , or . Therefore, the length of AC is and the length of BC is 103 . It’s given that angle ACB is a right angle. Therefore, triangle ACB is a right triangle with legs AC and BC and hypotenuse AB. By the Pythagorean theorem, if a right triangle has a hypotenuse with length c and legs with lengths a and b, 2+ 2= 2 103 then a b c . Substituting for a and b in this equation yields 103 2 + 103 2 = 2 103+103= 2 206= 2 ^ h ^ h c , or c , which is equivalent to c . Taking 206= the positive square root of both sides of this equation yields c. Therefore, 206 the length of AB is . Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This would be the length of AB if the length of AC were 103 103 , not . Choice D is incorrect and may result from conceptual or calculation errors.