AMSCO'S Geometry. New York - Rye High School

600 Geometry of the CircleIn this exploration, we will construct a regular triangle (equilateral triangle),a regular hexagon, a regular quadrilateral (a square), a regular octagon, and aregular dodecagon (a polygon with 12 sides) inscribed in a circle.a. Explain how a compass and a straightedge can be used to construct an equilateraltriangle. Prove that your construction is valid.b. Explain how the construction in part a can be used to construct a regularhexagon. Prove that your construction is valid.c. Explain how a square, that is, a regular quadrilateral, can be inscribed in acircle using only a compass and a straightedge. (Hint: What is true about thediagonals of a square?) Prove that your construction is valid.d. Bisect the arcs determined by the chords that are sides of the square fromthe construction in part c. Join the endpoints of the chords that are formedto draw a regular octagon. Prove that this construction is valid.e. A regular octagon can also be constructed by constructing eight isoscelestriangles. The interior angles of a regular octagon measure 135 degrees.Bisect a right angle to construct an angle of 45 degrees. The complement ofthat angle is an angle of 135 degrees. Bisect this angle to construct the baseangle of the isosceles triangles needed to construct a regular octagon.f. Explain how a regular hexagon can be inscribed in a circle using only a compassand a straightedge. (Hint: Recall how a regular polygon can be dividedinto congruent isosceles triangles.)g. Bisect the arcs determined by the chords that are sides of the hexagon frompart f to draw a regular dodecagon.h. A regular dodecagon can also be constructed by constructing twelve isoscelestriangles. The interior angles of a regular dodecagon measure 150degrees. Bisect a 60-degree angle to construct an angle of 30 degrees. Thecomplement of that angle is an angle of 150 degrees. Bisect this angle to constructthe base angle of the isosceles triangles needed to construct a regulardodecagon.CUMULATIVE REVIEW Chapters 1–13Part IAnswer all questions in this part. Each correct answer will receive 2 credits. Nopartial credit will be allowed.1. The measure of A is 12 degrees more than twice the measure of its complement.The measure of A is(1) 26 (2) 39 (3) 64 (4) 124