32. The figure shows a circle with center P and inscribed isosceles triangle ABC.
This question is not difficult if we remember the construction of a regular hexagon. We found out that a chord congruent to a radius subtends an arc of measure 60 degrees -- a sixth of the circumference, which explains why it leads to a regular hexagon. Then angle ABC, as an inscribed angle, would measure half of the subtended arc, or 30 degrees.
Notice that we made an assumption here -- in isosceles triangle ABC,
Other than that, this should be a simple problem. I'm not sure what the most common student error will be -- 60, maybe, or otherwise the student doesn't know how to solve this problem at all.
PARCC Practice EOY Question 32
U of Chicago Correspondence: Lesson 15-3, The Inscribed Angle Theorem
Key Theorem: Inscribed Angle Theorem
In a circle, the measure of an inscribed arc is one-half the measure of its intercepted arc.
Common Core Standard:
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Commentary: Today is our final activity day. I just couldn't resist having another construction activity, to construct the figure in the PARCC problem, which is a 30-75-75 triangle inscribed in a circle. To construct the figure, we start out just as we would for a regular hexagon -- draw a circle, mark a point A, then without changing the radius, mark another point C. The easiest way to locate B is to construct the perpendicular bisector of AC. We can take advantage of the fact that we already have a point on the perpendicular bisector of