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.
9. In right triangle ABC below, m∠C = 90°, AC = 12, and m∠A = 25°.
Which equation is correct for nABC?
(1) a = 12 / tan 25°
(2) a = 12 tan 25°
(3) c = 12 / tan 25°
(4) c = 12 tan 25°
Answer: (2) a = 12 tan 25°
Side c is the hypotenuse and opposite from angle C. The hypotenuse is not used for the tangent function, so eliminate Choices (3) and (4) immediately.
Tan A = opp/adj = a / 12
Therefore, a = 12 tan A = 12 tan 25°, which is Choice (2).
10. Triangle HUS is shown below.
If point G is located on US and HG is drawn, which additional information is sufficient to prove △HUG ≅ △HSG by SAS?
(1) HG bisects US
(2) HG is an altitude
(3) HG bisects ∠UHS
(4) HG is the perpendicular bisector of US.
Answer: (4) HG is the perpendicular bisector of US.
For SAS to apply, you would need two pairs of corresponding sides and the pair of angles between them to be congruent.
In Choice (1), HG bisecting US would give us two pairs of congruent sides, but it doesn't guarantee that the angles are congruent. Eliminate Choice (1).
In Choice (2), HG being an altitude would mean that each triangle would have a right angle and contain side HG, but we wouldn't know that SG is congruent to GU. Eliminate Choice (2).
In Choice (3), HG bisecting ∠UHS would mean that each triangle would have a congruent angle and contain side HG, but we wouldn't anything about sides HU and HS. Eliminate Choice (3).
In Choice (4), HG bisecting US perpendicularly would give us two pairs of congruent sides and a pair of right angles. So △HUG ≅ △HSG by SAS. Choice (4) is the correct answer.
11. The area of the base of a cone is 9π square inches. The volume of the
cone is 36π cubic inches. What is the height of the cone in inches?
(1) 12
(2) 8
(3) 3
(4) 4
Answer: (1) 12
The Volume of a cone is 1/3 of the Area of the circular base times its height.
36π = (1/3) (9π) h
36π = (3π) h
12 = h
The correct answer is Choice (1).
If the problem was about a cylinder instead of a cone, then (4) would've been the answer.
12. On the set of axes below, AB, CD, EF, GH, and IJ are drawn.
Which segment is the image of AB after a dilation with a scale factor of 2 centered at (-2,-1)?
(1) CD
(2) EF
(3) GH
(4) IJ
Answer: (2) EF
Starting at point (-2,-1), you have to go 3 units up to get to (-2,2), which is a point on AB. If point (-2,2) is dilated with a scale factor of 2 centered on (-2,-1), it will be another 3 units up, which is the point at (-2,5), which is a point on line EF.
If you want more confirmation, consider that to get from (-2,1) to point B, you have to go 2 units up and 4 units to the right. Starting at point B, it you go another 2 units up and 4 units to the right, you end up at point F.
The correct answer is Choice (2).
13. Trapezoid ABCD is graphed on the set of axes below.
Which transformation would map point A onto A'(3,-7)?
(1) reflection over y = x
(2) reflection over the y-axis
(3) rotation of 180° about (0,0)
(4) rotation of 90° counterclockwise about (0,0)
Answer: (1) reflection over y = x
Point A is located at (-7,3) in Quadrant II. Point A' will be at (3,-7) is quadrant IV.
In Choice (1), a reflection over the line y = x would switch the x and y coordinates so that (-7,3) goes to (3,-7). This is the correct answer.
In Choice (2), a reflection over the y-axis would bring (-7,3) to (7,3). Eliminate Choice (2).
In Choice (3), a rotation of 180° about (0,0) would bring (-7,3) to (3,-7). Eliminate Choice (3).
In Choice (4), rotation of 90° counterclockwise about (0,0) would bring (-7,3) into Quadrant IV at point (-3,-7). Eliminate Choice (4).
14. A storage building is modeled below by a hemisphere on top of a cylinder. The diameter of both the cylinder and hemisphere is 12 feet. The total height of the storage building is 30 feet.
To the nearest cubic foot, what is the volume of the storage building?
(1) 942
(2) 2488
(3) 3167
(4) 3845
Answer: (3) 3167
If the diameter of the hemisphere is 12 feet, then the radius of the hemisphere is 6 feet. This means that the height of the cylinder is 30 - 6 = 24 feet. The volume of the building equals the volume of the cylinder plus the volume of the hemisphere.
V = πr2h + (1/2) (4/3) πr3
= π (6)2 (24) + (2/3) π (6)3 = 3166.7
That's 3167 to the nearest cubic foot, which is Choice (3).
Side note: in Choice (4), you would get 3845 if you used 30 as the height of the cylinder instead of 24. I don't have a guess for how they came up with the numbers in Choices (1) and (2).
15. Which regular polygon will carry onto itself after a 135° rotation about
its center?
(1) triangle
(2) pentagon
(3) hexagon
(4) octagon
Answer: (4) octagon
Which of the listed polygons have an exterior angle that is a factor of 135°?
A regular triangle is equilateral and has 360°/3 = 120° exterior angles, which is not a factor of 135°.
A regular pentagon has 360°/5 = 72° angles, which is not a factor of 135°.
A regular hexagon has 360°/6 = 60° angles, which is not a factor of 135°.
And a regular pentagon has 360°/8 = 45° angles, which is a factor of 135°.
Choice (4) is correct.
16. What is the length of the radius of the circle whose equation is
x2 + y2 - 2x + 4y - 5 = 0?
(1) √(5)
(2) √(10)
(3) 5
(4) 10
Answer: (2) √(10)
Rearrange the terms and complete the squares to rewrite the equation in standard form.
x2 + y2 - 2x + 4y - 5 = 0
x2 - 2x + y2 + 4y - 5 = 0
x2 - 2x + 1 + y2 + 4y + 4 - 5 + 5 = 0 + 1 + 4 + 5
(x - 1)2 + (y - 2)2 = 10
So the square of the length of the radius, r2 = 10. That means that r = √(10), which is Choice (2).
More to come. Comments and questions welcome.
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