Geometry Problems of the Day (Geometry Regents, June 2012)

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 2012

Part I: Each correct answer will receive 2 credits.


25. In the diagram below of triangle ABC, BC is extended to D.

If m∠A = x² - 6x, m∠B = 2x - 3, and m∠ACD = 9x + 27, what is the value of x?

1) 10
2) 2
3) 3
4) 15

Answer: 4) 15

The Remote Angle Theorem states that the sum of Angle A and Angle B must equal the measure of angle ACD. If you didn't know this, you would have to substitute each value for x to get A and B and then calculate ACB to see if you got a linear pair. It would take a while, but it would work.

x² - 6x + 2x - 3 = 9x + 27
x² - 13x - 30 = 0
(x - 15)(x + 2) = 0
x - 15 = 0 or x + 2 = 0
x = 15 or x = -2
Discard the negative result.

Check: (15)² - 6(15) = 135
2(15) - 3 = 27
9(15) + 27 = 162

135 + 27 = 162 (check!)

26. An equation of the line that passes through (2,-1) and is parallel to the line 2y + 3x = 8 is

1) y = 3/2 x - 4
2) y = 3/2 x + 4
3) y = -3/2 x - 2
4) y = -3/2 x + 2

Answer: 4) y = -3/2 x + 2

A line that is parallel to the given line will have the same slope as the given line. So first thing is to find the slope of the given line.

In Standard Form Ax + By = C, the slope is given by -A/B, which is -3/2. Note that the order of the terms in the question is reversed with By first. Eliminate Choices (1) and (2).

If you needed to rewrite the equation in slope-intercept form:

2y + 3x = 8
2y = - 3x + 8
y = -3/2 x + 4

Check the point (2,-1) to see if it is a solution to either line:

y = -3/2(2) - 2 = -3 - 2 = -5, not -1. Eliminate Choice (3).

y = -3/2(2) + 2 = -3 + 2 = -1. This is the correct answer.

27. The graph below shows JT and its image, J'T', after a transformation.

What transformation would map JT onto J'T'?

1) translation
2) glide reflection
3) rotation centered at the origin
4) reflection through the origin

Answer: 2) glide reflection

JT is reflected over the x-axis and then a translation moves it to the right. This is a glide reflection.

Choice (1) is incorrect because there is no one translation that could map this image.

Choice (3) is incorrect because T' would be at point (2,-4) not (4,-4).

Choice (4) is actually the same thing as Choice (3), so both choices could be eliminated because they cannot both be correct.

28. Which reason could be used to prove that a parallelogram is a rhombus?

1) Diagonals are congruent.
2) Opposite sides are parallel.
3) Diagonals are perpendicular.
4) Opposite angles are congruent.

Answer: 3) Diagonals are perpendicular.

The usual tests for a rhombus are that all four sides as congruent and the diagonals are perpendicular.

Choice (1) is true for rectangles (and also isosceles trapezoids, which aren't parallelograms).Eliminate (1).

Choice (2) is true for all parallelograms. Eliminate (2).

Choice (4) is true for all parallelograms. Eliminate (4).

End of Part I.

More to come. Comments and questions welcome.

More Regents problems.

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