Geometry Problems of the Day (Geometry Regents, August 2010)

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 2010

Part II: Each correct answer will receive 2 credits. Partial credit is available. One computational or conceptual error will cost 1 credit. A second will cost the other.


32. On the line segment below, use a compass and straightedge to construct equilateral triangle ABC. [Leave all construction marks.]

Answer:

Place the point of the compass on point A and open it up until the pencil is on point B.

Make an arc above the line. Without changing the size of the compass, turn the compass around and make a second arc from point A. Make a point where the arcs intersect. Use the straightedge to draw the triangle.

Look at the image below. Do not erase any of the marks. Do not add any others. If you don't use the compass or the straightedge, it will be obvious that you did not and you will lose credit.

33. AIn the diagram below, car A is parked 7 miles from car B. Sketch the points that are 4 miles from car A and sketch the points that are 4 miles from car B. Label with an X all points that satisfy both conditions.

Answer:

All the points that are 4 miles from point A is a circle with point A at the center. Likewise, you need to draw a circle around point B.

Because the points are only 7 miles apart, the two radii will overlap because 4 + 4 = 8, which is more than 7.

Your "sketch" doesn't have to be perfect but your circles must intersect at two points in between A and B. The circles should be about the same size, the best that you can draw them that way. (Again, it doesn't have to be perfect.) Points A and B should not be together inside either circle.

34. Write an equation for circle O shown on the graph below.

Answer:

The equation for a circle is (x - h)² + (y - k)² = r², where (h,k) is the center of the circle, and r is the radius. Notice that there are minus signs in the equation.

The center of the circle is (-1,2) and the radius is 6. (Start at point O and count the boxes until you hit the circle.)

The equation is (x - (-1))² + (y - 2)² = 6²

which is (x + 1)² + (y - 2)² = 36

End of Part II.

More to come. Comments and questions welcome.

More Regents problems.

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