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 I: Each correct answer will receive 2 credits.


1. Plane P is parallel to plane Q. If plane P is perpendicular to line ℓ, then plane Q

1) contains line ℓ
2) is parallel to line ℓ
3) is perpendicular to line ℓ
4) intersects, but is not perpendicular to line ℓ

Answer: 3) is perpendicular to line ℓ

If two planes are parallel to each other and a line is perpendicular to one of them, it must be parallel to the other.

Think of a line on the wall that runs from the floor to the ceiling, or the leg of a table that is perpendicular to the floor. Or even a pencil between two sheets of paper.

2. In the diagram below, quadrilateral ABCD has vertices A(-5,1), B(6,-1), C(3,5), and D(-2,7).

What are the coordinates of the midpoint of diagonal AC?
1) (-1, 3)
2) (1, 3)
3) (1, 4)
4) (2, 3)

Answer: 1) (-1, 3)

If you draw the line connecting them, you will see that you will move 8 boxes along the x axis and 4 boxes along the y axis. That means the midpoint is the point that is 4 units more than the x-coordinate of point A (and 4 less than that of C) and it will be 2 units more than the y-coordinate of point A (and 2 less than that of C).

Midpoint = (-5 + 4, 1 + 2) = (-1, 3).

3. In the diagram below, transversal intersects and at V and W, respectively.

If m∠TVQ = 5x - 22 and m∠VWS = 3x + 10, for which value of x is PQ || RS?
1) 6
2) 16
3) 24
4) 28

Answer: 2) 16

Angles TVQ and VWS are corresponding angles. If PQ || RS then the corresponding angles would be congruent. Conversely, if the corresponding angles are congruent, then the lines are parallel.

Set up an equation that says that 5x - 22 = 3x + 10

5x - 22 = 3x + 10
2x - 22 = 10
2x = 32
x = 16

If you had written 5x - 22 + 3x + 10 = 180, you would have gotten x = 24, which is Choice (3), and you wouldn't have known you made a mistake.

If you had written 5x - 22 + 3x + 10 = 90, you would have gotten x = 51/4, which isn't one of the choices, so you would've been lucky.

I'm surprised 4 isn't an option because 8x = 32 (one computational error) would give x = 4 as a result.

4. The measures of the angles of a triangle are in the ratio 2:3:4. In degrees, the measure of the largest angle of the triangle is
1) 20
2) 40
3) 80
4) 100

Answer: 3) 80

Common sense should tell you that the largest anlge of a triangle cannot be 20 or 40. Eliminate these two Choices. The least amount that the largest angle can be is 60, which would be an equilateral triangle. (And you could argue that there is no "largest" angle in an equilateral triangle, but that is outside the scope of this question.)

Set up an equation using multiples of x for each value in the ratio.

2x + 3x + 4x = 180
9x = 180
x = 20
The largest angle is 4x, which is 4(20) = 80.

5.The diameter of the base of a right circular cylinder is 6 cm and its height is 15 cm. In square centimeters, the lateral area of the cylinder is

1) 180π
2) 135π
3) 90π
4) 45π

Answer: 3) 90π

The lateral area is the surface area of the rectangle around the cylinder without the two circles on the top and bottom. Think of it as the label on a soup can or a can of vegetables.

A = L * W, but the length is the circumference of the circle and the width is the height of the cylinder, so:

A = π D h = π (6) (15) = 90π

More to come. Comments and questions welcome.

More Regents problems.