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 2013
Part I: Each correct answer will receive 2 credits.
16. In a coordinate plane, the locus of points 5 units from the x-axis is the
1) lines x = 5 and x = -5
2) lines y = 5 and y = -5
3) line x = 5, only
4) line y = 5, only
Answer: 2) lines y = 5 and y = -5
The x-axis is the line y = 0.
The locus of points 5 units away from y = 0 are parallel lines 5 units above it and below it.
That would be y = -5 and y = 5.
17.The sides of a triangle are 8, 12, and 15. The longest side of a similar triangle is 18. What is the ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle?
1) 2:3
2) 4:9
3) 5:6
4) 25:36
Answer: 3) 5:6
Perimeter has the same scale as the corresponding sides do.
The scale factor is 15/18, which simplifies to 5/6.
Choice (4) would be correct for Area.
Choice (1) would be correct if the question asked about the middle side instead of the largest side, with Choice (2) being about the area using the middle sides.
18. Lines m and n are in plane A. What is the converse of the statement
“If lines m and n are parallel, then lines m and n do not intersect”?
1) If lines m and n are not parallel, then lines m and n intersect.
2) If lines m and n are not parallel, then lines m and n do not intersect
3) If lines m and n intersect, then lines m and n are not parallel.
4) If lines m and n do not intersect, then lines m and n are parallel.
Answer: 4) If lines m and n do not intersect, then lines m and n are parallel.
The converse is the statment that is formed when you switch the order of the c;auses in the original conditional.
Choice (1) is the Inverse.
Choice (2) is an incorrectly written Inverse. It is nothing is relation to the original.
Choice (3) is the Contrapositive, which is the Inverse and Converse combined.
19. When the system of equations y + 2 = (x - 4)2 and 2x + y - 6 = 0
is solved graphically, the solution is
1) (-4, -2) and (-2, 2)
2) (4, -2) and (2, 2)
3) (-4, 2) and (-6, 6)
4) (4, 2) and (6, 6)
Answer: 2) (4, -2) and (2, 2)
It's a multiple-choice question, so you can solve it any way you want. However, you can rewrite the equations, putting them into "y=" form so that you can enter them into your graphing calculator.
Or you can algebraically check the points in the choices to see which points are solutions.
The "easiest" point to check is (2, 2). I say "easiest" because they're small numbers.
2 + 2 = (x - 4)2
4 = (-2)2 = 4. Check. (2, 2) is a solution.
Only one choice has (2, 2) as a solution, so that is the answer.
20. In the diagram of UVW below, A is the midpoint of UV, B is the midpoint of UW, C is the midpoint of VW, and AB and AC are drawn.
If VW = 7x - 3 and AB = 3x + 1, what is the length of VC?
1) 5
2) 13
3) 16
4) 32
Answer: 3) 16
AB is a midsegment and is half the length of VW. VC is half the length of VW and is the same length as VC.
You can write and equation that takes half of VW, but that means fractions, or you can write one that doubles AB.
7x - 3 = 2(3x + 1)
7x - 3 = 6x + 2
x = 5
VC = AB = 3x + 1 = 3(5) + 1 = 16
Note that Choice (1) is x, not VC, and Choice (4) is VW, not VC.
More to come. Comments and questions welcome.
More Regents problems.
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