Geometry Problems of the Day (Geometry Regents, June 2011)

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 2011

Part I: Each correct answer will receive 2 credits.


25. Which quadrilateral has diagonals that always bisect its angles and also bisect each other?

1) rhombus
2) rectangle
3) parallelogram
4) isosceles trapezoid

Answer: 1) rhombus

Only the rhombus diagonals bisect the diagonals. Rectangles do not (unless they are squares, which are rhombuses). Isosceles trapezoid diagonals do not bisect each other.

26. When △ABC is dilated by a scale factor of 2, its image is △ A′B′C′. Which statement is true?

1) AC ≅ A′C′
2) ∠A ≅ ∠A′
3) perimeter of △ ABC = perimeter of △ A′B′C′
4) 2(area of △ABC) = area of △ A′B′C′

Answer: 2) ∠A ≅ ∠A′

A dilation will not change the shape, so the angles will be congruent.

The lengths of the sides will change. In this case, they are doubled in length. So Choice (1) is eliminated because the sides are not congruent.

If the sides aren't congruent, then the perimeters cannot be equal. The perimeter of the image is double. Eliminate Choice (3).

The image will have double the base and double the height, so it will have FOUR times the area, not two. Eliminate Choice (4).

27. What is the slope of a line that is perpendicular to the line whose equation is 3x + 5y = 4?

1) -3/5
2) 3/5
3) -5/3
4) 5/3

Answer: 4) 5/3

Perpendicular lines have negative reciprocal slopes. That is, there product is -1. The exception is vertical lines are perpendicular to horizontal lines.

To find the slope perpendicular to the given line, you need the slope of the given line. When a linear equation is written in standard form, Ax + By = C, the slope is just -A/B. If you didn't remember this, you could have rewritten it into slope-intercept form: y = -3/5 x + 4/5

Since the slope of the line is -3/5, then the perpendicular slope must be +5/3 because (-3/5)(5/3) = -1.

28. In the diagram below of right triangle ABC, altitude BD is drawn to hypotenuse AC , AC = 16, and CD = 7.

What is the length of BD?

1) 3 √(7)
2) 4 √(7)
3) 7 √(3)
4) 12

Answer: 1) 3 √(7)

The Right Triangle Altitude Theorem tells us that (AD)(CD) = (BC)² because the triangles are similar and corresponding sides are proportional.

AD can be found by subtracting 16 - 7 = 9.

So (9)((7) = 63 = BC²

BC = √(9*7) = √9 * √7 = 3 √7, which is Choice (1).

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. Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.