Geometry Problems of the Day (Geometry Regents, January 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, January 2014

Part I: Each correct answer will receive 2 credits.


1. The midpoint of AB is M(4,2). If the coordinates of A are (6,-4), what are the coordinates of B?

1) (1, -3)
2) (2, 8)
3) (5, -1)
4) (14, 0)

Answer: 2) (2, 8)

Remember that you are working backward here. You are NOT finding the midpoint. You are finding the other endpoint.

To get from A to the midpoint, you have to subtract 2 from the x-coordinate. Therefore, to get to the other endpoint, you have to subtract another 2. Since 4 - 2 = 2, then the x-coordinate of the endpoint must be 2. There is only one possible choice.

To check: from A to the midpoint, you need to go up 6 units from -4 to 2. Add another 6 units, and the y-coordinate of the endpoint must be 8.

This would be obvious if you make a sketch on graph paper and count the boxes, which is a fine alternative if you don't want to deal with formulas and subtracting signed numbers.

2. Which diagram shows the construction of a 45° angle?

Answer: 3)

To construct a 45° angle, you have to start with a 90° angle (a perpendicular line) and then bisect it.

Choice (1) appears to be an equilateral triangle that had an angle bisected. That angle would be 30 degrees. Look at this diagram if for no other reason that it was the answer to a Part III construction problem on the June 2014 regents. (Had I realized this, I could've saved from trouble!)

3. What are the coordinates of the center and the length of the radius computations. of the circle whose equation is (x + 1)² + (y - 5)² = 16?

1) (1, -5) and 16
2) (-1, 5) and 16
3) (1, -5) and 4
4) (-1, 5) and 4

Answer: 4) (-1, 5) and 4

The equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle.

If r² = 16, then r = 4. Eliminate Choices (1) and (2).

Since the signs are "flipped" because of the minus signs in the formula, the center is (-1, 5).

4. If distinct planes R and S are both perpendicular to line ℓ, which statement must always be true?
1) Plane R is parallel to plane S
2) Plane R is perpendicular to plane S
3) Planes R and S and line ℓ are all parallel.
4) The intersection of planes R and S is perpendicular to line ℓ.

Answer: 1) Plane R is parallel to plane S

Consider a beam running from the floor to the ceiling. The beam is parallel to both. The floor and the ceiling are parallel.

Line ℓ can't be parallel to planes R and S. It is stated to be perpendicular. It cannot be both. Parallel means it would not intersect. Perpendicular intersects at a right angle. Eliminate Choice (3).

5. If triangle ABC and its image, triangle A'B'C', are graphed on a set of axes, triangle ≅ triangle A'B'C' under each transformation except

1) D₂
2) R₉₀
3) r_y=x
4) T_(-2,3)

Answer: 1) D₂

A Dilation is not a rigid motion. It does not preserve size. The image will be similar, but not congruent.

D₂ means the image will be twice the size (dilation with a scale factor of 2). It's location will depend upon the center of dilation.

R₉₀ is a Rotation of 90 degrees counterclockwise (unless otherwise stated). Turning the preimage doesn't affect its size or shape, only its orientation.

r_{y = x} is a reflection of the preimage over the line y = x. A reflection doesn't affect its size or shape, only its orientation.

T_(-2,3) is a translation of the preimage 2 units to the left and 3 units up. A translation doesn't affect its size, shape, or orientation.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon.