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 I: Each correct answer will receive 2 credits.


1. In the diagram below, △ABC ≅ △XYZ.

Which two statements identify corresponding congruent parts for these triangles?

1) AB ≅ XY and ∠C ≅ ∠Y
2) AB ≅ YZ and ∠C ≅ ∠X
3) BC ≅ XY and ∠A ≅ ∠Y
4) BC ≅ YZ and ∠A ≅ ∠X

Answer: 4) BC ≅ YZ and ∠A ≅ ∠X

AB ≅ XY and BC ≅ YZ. Eliminate Choices (2) and (3).

∠A ≅ ∠X, ∠B ≅ ∠Y and ∠C ≅ ∠Z. Elimate Choice (1).

This leaves Choice (4), which is the correct answer.

2. A support beam between the floor and ceiling of a house forms a 90° angle with the floor. The builder wants to make sure that the floor and ceiling are parallel. Which angle should the support beam form with the ceiling?

1) 45°
2) 60°
3) 90°
4) 180°

Answer: 3) 90°

I hate to be this way with any math question (NARRATOR: He loves it.), but this is a gimme of a question.

The support beam in a house should form a 90° angle with both the floor and the ceiling (which becomes the next floor).

Strictly speaking in Geometry terms, if two lines are parallel, then the same-side interior angles created by a transversal are supplementary. 90 is supplementary to 90 because 180 - 90 = 90.

3. In the diagram below, the vertices of △DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of △ABC is 36 cm.

What is the length, in centimeters, of EF?

1) 6
2) 12
3) 18
4) 4

Answer: 1) 6

The midsegments are half of the length of the sides of the triangle that they are parallel to.

The perimeter of the equilateral triangle is 36, so each side is 12. So each midsegment is 6, which is Choice (1).

4. What is the solution of the following system of equations?

y = (x + 3)² − 4
y = 2x + 5

1) (0,-4)
2) (-4,0)
3) (−4,−3) and (0,5)
4) (−3,−4) and (5,0)

Answer: 3) (−4,−3) and (0,5)

You can put both of these in your calculator. Or you check each of the choices.

The point (0,-4) is NOT the y-intercept of y = 2x + 5. Eliminate Choice (1).

The point (-4,0) is NOT on y = 2x + 5 because 2(-4) + 5 = -3.

Since (0,5) is the y-intertecept of y = 2x + 5, check it with the quadratic:

y = (0 + 3)² - 4 = 3² - 4 = 9 - 4 = 5 So (0,5) is also a solution to the system of equations. This means that Choice (3) is the correct answer.

Double check point (-4,-3):

y = 2(-4) + 5 = -8 + 5 = -3, check!
y = (-4 + 3)² - 4 = (-1)² - 4 = 1 - 4 = -3, check!

5. One step in a construction uses the endpoints of AB to create arcs with the same radii. The arcs intersect above and below the segment. What is the relationship of AB and the line connecting the points of intersection of these arcs?

1) collinear
2) congruent
3) parallel
4) perpendicular

Answer: 4) perpendicular

These are the instructions for constructing a perpendicular bisector of line segment AB. This is Choice (4).

More to come. Comments and questions welcome.

More Regents problems.

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