Question

determine whether the following statement is true or false. if it is false, give a counterexample. you can circumscribe a circle about any three points in a plane. choose the correct answer below. a. false. if the points are collinear, then the angle bisectors will be overlapping and thus, will intersect everywhere. therefore, a circumscribed circle wont be possible. b. false. if the points are collinear, then the perpendicular bisectors will be parallel and thus, wont intersect. therefore, a circumscribed circle wont be possible. c. true. you can connect the points into a triangle and construct a circumscribed circle using known methods. d. false. if the points are too far apart, a circumscribed circle wont be possible.

Explanation:

A circumscribed circle (circumcircle) for three points exists only if the points are non-collinear. For collinear points, the perpendicular bisectors of the segments connecting them are parallel, so they never intersect to form a circumcenter (the center of the circumscribed circle). Option A incorrectly references angle bisectors, option C is wrong as collinear points can't form a triangle for a circumcircle, and option D is incorrect because distance between points doesn't prevent a circumcircle if they are non-collinear.

Answer:

B. False. If the points are collinear, then the perpendicular bisectors will be parallel and thus, won't intersect. Therefore, a circumscribed circle won't be possible.