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.
1. In right triangle LMN below, LN = 8, MN = 15, and LM = 17.
If triangle LMN is translated such that it maps onto triangle XYZ,
which statement is always true?
(1) XY = 15
(2) YZ = 17
(3) m∠Z = 90°
(4) m∠X = 90°
Answer: (3) m∠Z = 90°
If LMN is translated on XYZ, then L maps to X, M maps to Y, and N maps to Z, and those pairs of angles are congruent. Also, LM maps to XY, MN maps to YZ, and LN maps to XZ.
Choice (1) says XY = 15, but LM = 17, so this is incorrect.
Choice (2) says YZ = 17, but MN = 15, so this is incorrect.
Choice (3) says Z is a right angle. N maps to Z, and N is a right angle, so this is true.
Choice (4) says X is a right angle, but L maps to X, and L is not a right angle, so this is incorrect.
Choice (3) is the correct choice.
2. Directed line segment KC has endpoints K(-4,-2) and C(1,8).
Point E divides KC such that KE:EC is 3:2. What are the coordinates of point E?
(1) (-1,4)
(2) (-2,2)
(3) (-3,0)
(4) (0,6)
Answer: (1) (-1,4)
The x-coordinate of E will be 3/5ths of the way from -4 to 1, and the y-coordinate of E will be 3/5ths of the way from -2 to 8.
From K to C, the x-coordinate changes 1 - (-4) = 5 units, and 3/5 of 5 = 3. Add 3 to -4 to get -1 as the x-coordinate of E.
From K to C, the y-coordinate changes 8 - (-2) = 10 units, and 3/5 of 10 = 6. Add 6 to -2 to get 4 as the y-coordinate of E.
Point E is at (-1, 4), which is Choice (1).
3. In right triangle DAN, m∠A = 90°. Which statement must always be
true?
(1) cos D = cos N
(2) cos D = sin N
(3) sin A = cos N
(4) cos A = tan N
Answer: (2) cos D = sin N
If A is the right angles, then the sin D = cos N and sin N = cos D.
Choice (2) is the correct answer.
4. In the diagram below of parallelogram RSTV, diagonals SV and RT intersect at E.
Which statement is always true?
(1) SR ≅ RV
(2) RT ≅ SV
(3) SE ≅ RE
(4) RE ≅ TE
Answer: (4) RE ≅ TE
The diagonals of a parallelogram bisect each other but they are not congruent to each other. So SE ≅ VE and RE ≅ TE.
Choice (1) has consecutive sides of a parallelogram, which are only congruent in rhombuses.
Choice (2) has congruent diagonals, which is only true in rectangles.
Choice (3) says that half of each diagonal must be congruent, but that is only true in rectangles.
Choice (4) says that the two halves of RT are congruent to each other, which is always true. Choice (4) is the correct answer.
5. In △SNA below, UE || NA.
If SU = 3, SN = 11, and EA = 13, what is the length of SE, to the nearest tenth?
(1) 2.5
(2) 3.5
(3) 4.9
(4) 17.9
Answer: (3) 4.9
Because UE || NA, SU / UN = SE / EA, so 3/8 = SE / 13.
Therefore, 8 SE = 39, and SE = 39/8, which is approximately 4.9. This is Choice (3).
6. Many roofs are slanted to prevent the buildup of snow. As modeled below, the length of a roof is 5.5 meters and it rises to a height of 2.5 meters
The angle of elevation of the roof, to the nearest degree, is
(1) 24°
(2) 25°
(3) 27°
(4) 28°
Answer: (3) 27°
You are given the hypotenuse of the triangle and the side that is opposite the angle of elevation. Opposite and hypotenuse means that you need to use the sine ratio.
sin x = 2.5 / 5.5
x = sin-1(2.5/5.5) = 27.03..., which is 27 to the nearest degree.
The correct answer is Choice (3).
As a side note, I'm surpised that they didn't include 63 degrees, which would be the result is you accidentally used sine. If you used tangent, you would have gotten 24.4, which rounds down to 24 or (mistakely) up to 25.
7. In the diagram below, CT || AR , and ACE and RC are drawn such that AC ≅ RC
If m∠ECT = 75°, what is m∠ACR?
(1) 30°
(2) 60°
(3) 75°
(4) 105°
Answer: (1) 30°
Triangle CAR is an isosceles triangle, so the base angles, CAR and ARC, are congruent. Angles ECT and CAR are corresponding angles, which are also congruent.
If angle ECT has a measure of 75 degrees, then CAR and ARC are each 75 degrees. This means that the measure of ACR, the third angle of triangle CAR, has a measure if 180 - (75 + 75) = 30 degrees.
This is Choice (1).
8. In the diagram below, △ABC has medians AX, BY, and CZ that
intersect at point P.
If AB = 26, AC = 28, and PC = 16, what is the perimeter of △CZA?
(1) 57
(2) 65
(3) 70
(4) 73
Answer: (2) 65
Medians are drawn to midpoints of the opposite sides, and the midpoint divides a line segment into two equal, congruent segments. Medians meet at a point called the centrod which is 2/3 of the way from the angle to the midpoint.
The perimeter of CZA is the sum of the sides CZ, ZA and AC.
CZ is the sum of CP + PZ, and CP is twice the length of PZ, so CZ = 16 + 8 = 24.
ZA is half the length of AB, which is 26, so ZA = 13.
AC is given to be 28.
Therefore, the perimeter of CZA is 24 + 13 + 28 = 65, which is Choice (2).
More to come. Comments and questions welcome.
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