Geometry Problems of the Day (Geometry Regents, August 2011)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, August 2011

Part II: Each correct answer will receive 2 credits. Partial credit is available.


29. In the diagram below, trapezoid ABCD, with bases AB and DC, is inscribed in circle O, with diameter DC. If m AB = 80, find m BC.

Answer:

Parallel chords in a circle intercept arcs which will be congruent. So arc AD ≅ BC. Arc CD is a semicircle, which is 180 degrees.

So x + 80 + x = 180

2x + 80 = 180

2x = 100

x = 50

The measure of arc BC is 50 degrees.

30. On the diagram of △ABC shown below, use a compass and straightedge to construct the perpendicular bisector of AC. [Leave all construction marks.]

Answer:

You want the perpendicular bisector of AC. The rest of the triangle is just a distraction. Ignore it.

To find a perpendicular bisector, center you compass on point A and open it more than halfway to C. Make two arcs, one above the line and one below. (You can also just make one really big arc -- no one will mind.) Repeat the process, centering the compass on point C without chaning the opening of the compass. Make two more arcs so that you have to little Xs.

With a straightedge, draw a line connecting the Xs. That is the perpendicular bisector.

31. A sphere has a diameter of 18 meters. Find the volume of the sphere, in cubic meters, in terms of π.

Answer:

Find the radius from the diameter and use the formula from the back of the booklet: V = 4/3 πr³.

V = (4/3)(π)(9) = 972π

More to come. Comments and questions welcome.

More Regents problems.

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