This exam was adminstered in August 2024.
More Regents problems.
August 2024 Geometry Regents
Part I
Each correct answer will receive 2 credits. No partial credit.
17. The line represented by the equation y = 4x + 15 is dilated by a
scale factor of 2 centered at the origin. Which equation represents its
image?
(1) y = 4x + 15
(2) y = 4x + 30
(3) y = 8x + 15
(4) y = 8x + 30
Answer: (2) y = 4x + 30
The slope of the line will not change because in a dilation the orientation of the image is the same as the pre-image. However, the y-intercept will be twice as far from the origin as the origin line.
Since the original line has a point at (0,15), the image will go through the point (0,30).
Choice (2) is the correct answer.
18. Line segment RH has endpoints R(-4,4) and H(2,-4). Which equation represents a line perpendicular to RH that passes through
the point (3,-1)?
(1) y + 1 = 3/4(x - 3)
(2) y + 1 = -3/4(x - 3)
(3) y + 1 = 4/3(x - 3)
(4) y + 1 = -4/3(x - 3)
Answer: (1) y + 1 = 3/4(x - 3)
The only difference in the choices is the slope of the line. So you need to find the slope of the original line and then find the slope of the line that is perpendicular to it.
The slope of line RH is △y/△x = the change in y / the change in x = (-4 - 4) / (2 - (-4)) = -8 / 6 = -4/3.
The slope of a line perpendicular to RH would have a slope that was the inverse reciprocal of the slope of RH, which would be +3/4. This is Choice (1).
Check: (-4/3)(+3/4) = (-12/12) = -1. The lines are perpendicualr.
19. In right triangle SNO below, altitude NW is drawn to hypotenuse SO.
Which statement is not always true?
(1) SO/SN = SN/SW
(2) SW/NS = NS/OW
(3) SO/ON = ON/OW
(4) OW/NW = NW/SW
Answer: (2) SW/NS = NS/OW
There are 3 right triangles in the image: the large one and the two smaller one. All three triangles are similar, meaning that they have the same shape, congruent corresponding angles, and corresponding sides that are proportional. Each triangle has a hypotneuse across from the right angle, and a longer leg and a shorter leg across from the larger and smaller acute angles, respectively.
The proportion in Choice (1) compares the hypotenuse / short leg of SON with the hypotenuse / short leg of NSW. Eliminate Choice (1).
The proportion in Choice (2) compares the short leg / hypotenuse NSW with two legs of two different triangles. There is no reason for these to ratios to be equal to each other. Choice (2) is the correct answer.
The proportion in Choice (3) compares the hypotenuse / long leg of SON with the hypotenuse / long leg of NOW. Elimnate Choice (3).
The proportion in Choice (4) compares the long leg / short leg of NOW with the long leg / short leg of NSW. Eliminate Choice (4).
20. A rectangle has a width of 3 and a length of 4. The rectangle is dilated by a scale factor of 1.8. What is the area of its image, to the nearest tenth?
(1) 3.7
(2) 6.7
(3) 21.6
(4) 38.9
Answer: (4) 38.9
The area of the image is Length X scale factor times Width X scale factor.
A = 3 * 1.8 * 4 * 1.8 = 38.88 = 38.9
The correct answer is Choice (4).
Notice that the original rectangle had an area = 3 * 4 = 12, and the image was bigger, so Choices (1) and (2) are just silly. Choice (3) is the product of 3 * 4 * 1.8.
21. In the diagram below of circle P, diameter MD and chord AL intersect at Q, m∠AQD = 46°, and mLD = 124°.
What is m AD?
(1) 36°
(2) 46°
(3) 51°
(4) 92°
Answer: (1) 36°
You have to work your way around the circle to fill in other things you know. For example, since MD is a diameter, then we know that the sum of arcs ML and LD is 180 degrees. Since LD = 124, then ML = 180 - 124 = 56 degrees.
The sum of arc ML and AD is equal to twice the size of angle AQD, so 56 + AD = 92, and AD = 92 - 56 = 36 degrees, which is Choice (1).
Logically, since Q is not the center of the circle, arc AD was going to be less than 46 degrees, and there was only one choice that made since.
The rule that was used above says, basically, that if two intersecting chords intercept two arcs, the vertical angles were the chords intersect will be equal to the average of the two arc sizes. In this case, the arcs are 56 and 36 degrees, which average to 46 degrees.
22. The right prism with a triangular base shown below is cut by a plane perpendicular to its bases.
The two-dimensional shape of the cross section is always a
(1) triangle
(2) rhombus
(3) pentagon
(4) rectangle
Answer: (4) rectangle
Perpendicular to the bases means you're cutting up and down like you're slicing into a wedge of cheese. When you cut it that way, no matter how you slice it, the cross section will be a rectangle composed of a line from the top and bottom planes (parallel to each other) and two vertical lines down two of the three sides of the triangle. Those two vertical lines are also parallel. Two pairs of parallel sides make a parallelogram, but the sides are horizontal and vertical, making right angles, so it's a rectangle.
The correct answer is Choice (4).
If the question had said parallel to the bases, slicing horizontally between the top and bottom bases, then the cross section would have been a triangle.
The other two choices, pentagon and rhombus, are just silly. Why not an octagon?
23. A rectangular fish tank measures 24 inches long, 12 inches wide, and 16 inches high, as modeled in the diagram below.
If the empty tank weighs 25 pounds and the fish tank is filled with water to a height of 14 inches, what is the approximate weight of the tank and water?
[27.7 in3 = 1 pound of water]
(1) 146
(2) 166
(3) 171
(4) 191
Answer: (3) 171
First of all, the height of the tank doesn't matter because there's only 14 inches of water in the tank, so that's the "height" that we need. Second, you have to find the Volume in cubic inches and then convert that to pounds of water.
Volume = L X W X H = (24)(12)(14) = 4032 in3
Weight of the water = (4032 in3) / (27.7 in3 / pound) = 145.56 pounds
Weight of the tank = 25 pounds
Total weight = 146 + 25 = 171, which is Choice (3).
Notice that 146 was one of the incorrect Choices.
24. A circle has a radius of 4.5. What is the measure of the central angle
that intercepts an arc whose length is 6.2, to the nearest degree?
(1) 35°
(2) 42°
(3) 64°
(4) 79°
Answer: (4) 79°
Find the circumference of the circle and then find the part of the circle that the arc represents.
C = 2 (3.141592)(4.5) = 28.274328
Divide 6.2/28.274328 = 0.219
Multiply 0.219 * 360° = 78.84° or 79°, which is Choice (4).
End of Part I.
How did you do?
Comments and questions welcome.
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