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, January 2012
Part I: Each correct answer will receive 2 credits.
11. When a quadrilateral is reflected over the line y = x, which geometric
relationship is not preserved?
1) congruence
2) orientation
3) parallelism
4) perpendicularlity
Answer: 2) orientation
If you flip something over a diagonal line then the orientation is not going to be preserved.
You can draw a quick image, something other than a square, and try it yourself. It will be facing a different way after the reflection.
The pre-image and the image will be congruent, having the same shape and size. Lines will still be parallel or perpendicular. None of that changes. Just the direction it's pointed in.
I'm surprised the question wasn't more specific because if you did sketch a square, you might not have gotten the answer right away.
12. Which equation represents circle O with center (2, -8) and radius 9?
1)(x + 2)2 + (y - 8)2 = 9
2) (x - 2)2 + (y + 8)2 = 9
3) (x + 2)2 + (y - 8)2 = 81
4) (x - 2)2 + (y + 8)2 = 81
Answer: 4) (x - 2)2 + (y + 8)2 = 81
The equation of a circle is given by the formula (x - h)2 + (y - k)2 = r2, where (h,k) in the center of the circle and r is the radius. Note that there are MINUS signs in the formula, so the signs will be flipped.
Since the radius is 9, then r2 = 81. Eliminate Choices (1) and (2).
Since the signs are flipped, the center (2,-8) means that we want (x - 2) and (y + 8). This is Choice (4).
13. Which statement is the negation of “Two is a prime number” and
what is the truth value of the negation?
If AB = 20, AE = 12, and CF = 15, what is the length of AC?
1) Two is not a prime number; false
2) Two is not a prime number; true
3) A prime number is two; false
4) A prime number is two; true
Answer: 1) Two is not a prime number; false
To negate a statement, you have to add a "not", or remove a "not" that is already there.
The negation of "Two is a prime number" is "Two is not a prime number." However, since 2 actually is a prime number, the negation is false.
14. In the diagram below of circle O, chords AB and CD intersect at E.
If m∠AEC = 34 and mAC = 50, what is mDB ?
1) 16
2) 18
3) 68
4) 118
Answer: 2) 18
Subtract 50 - 34 = 16. Then subtract 34 - 16 = 18. The answer is Choice (2).
When two chords intersect, they form vertical angles. They also intercept two pairs of arcs. The average of two opposite arcs will be the size of the vertical angle in the middle.
In other words, the average of arcs AC and BD will be the same angles AEC and BED.
You could solve it algebraically using the equation:
50 + x = 68
x = 18
Or you can realize that an average of two numbers can be found through leveling to find the middle number, which you could invert if you have the middle number and need either the higher or lower one.
If you scan the above text, you can see how two of the incorrect choices were arrived at. Choice (4) seems to be the sum of 50 and 68.
15. The volume of a rectangular prism is 144 cubic inches. The height
of the prism is 8 inches. Which measurements, in inches, could be
the dimensions of the base?
1) 3.3 by 5.5
2) 2.5 by 7.2
3) 12 by 8
4) 9 by 9
Answer: 2) 2.5 by 7.2
Use the formula V = L * W * H, and substitute what you are given.
144 = L * W * 8, so LW = 16. So the product of the dimensions must be 16. Obviously, Choices (3) and (4) are too big.
Choice (1), 3.3 * 5.5 = 18.15
Choice (2), 2.5 * 7.2 = 18
Once you eliminated Choices (3) and (4), it might have been obvious that Choice (1) would not have an integer answer, multiplying .3 times .5.
More to come. Comments and questions welcome.
More Regents problems.
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