Geometry Problems of the Day (Geometry Regents, June 2014)

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, June 2014

Part III: Each correct answer will receive 4 credits. Partial credit is possible.


35. The graph below shows triangle A'B'C', the image of ABC after it was reflected over the y-axis.

Graph and label ABC, the pre-image of A'B'C'.

Graph and label A"B"C", the image of A'B'C' after it is reflected through the origin.

State a single transformation that will map ABC onto A"B"C".

Answer:

First, you must reflect A'B'C' over the y-axis to get the pre-image (which was reflected over the y-axis to get A'B'C'). It will be in Quadrant II.

Next, reflect A'B'C' about the origin to get A"B"C", which will be in Quadrant III. Reflecting over the origin will be the same as rotating the image 180 degrees.

Do this before answering the final question.

A single transformation of r_x-axis, a reflection over the x-axis, would map triangle ABC onto triangle A"B"C".

36.On the set of axes below, sketch the locus of points 2 units from the x-axis and sketch the locus of points 6 units from the point (0,4). Label with an X all points that satisfy both conditions.

Answer:

The locus of points 2 units away from the x-axis would be two lines parallel to the x-axis, one above it, and one below it. (Make sure you do the x-axis and not the y-axis.)

The locus of points 6 units from a point will form a circle with a raidus of 6. You can sketch it as best as you can. Start with (0, 4) and mark points (6, 4), (-6, 4), (0, 10) and (0, -2). Make the best circle that you can. It doesn't have to be perfect, but don't let it look like a diamond or an octagon or something crazy.

When you are done with the circle, you will see where the circle intersects to the two parallel lines. Mark those 3 points off with Xs.

37. 7 Using a compass and straightedge, construct an equilateral triangle with AB as a side.

Using this triangle, construct a 30° angle with its vertex at A.
[Leave all construction marks.]

Answer:

Constructing an equilateral triangle from a line segment isn't difficult. Using the compass, measure the length from point A to point B. Now swing the compass up and make an arc above the center of the line segment.

Turn the compass around without changing its size, and place the end on B. Make an arc above the line. It will intersect the first arc you drew. Label the intersection point C.

Use the straightedge to draw AC and BC. The first part is done.

To construct a 30 degree angle isn't difficult because you already have a 60 degree angle in the equilateral triangle. You just have to bisect it.

To bisect the angle, put the compass at A and make a small arc that intersects AC and AB. I'll call these D and E just for clarity, but you don't need to label these. From D make an arc in the center of the triangle. From E make the same size arc so that it intersects with the one you just made. You could call this F. Use the straightedge to draw AF. You made a 30 degree angle at vertex A.

You final image might look something like this, only a little better I hope. I can't use a compass and straightedge in Windows Paint.

UPDATE: Here is a better constructed example which appeared as an incorrect answer to a multiple-choice problem on the January 2014 Geometry Regents!

End of Part III.

More to come. Comments and questions welcome.

More Regents problems.