Wednesday, July 05, 2017

June 2017: Common Core Geometry Regents, Part 1

The following are the questions and answers (and commentary) for part of the New York State Algebra Regents exam. If you have any questions or comments (or corrections), please add them in the Comments section.

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

Answers to Part II can be found here.

Answers to Part III can be found here.

Answers to Part IV can be found here.

June 2017, Geometry (Common Core), Part I

1.In the diagram below, triangle ABC = triangle DEF. (image omitted)
Which sequence of transformations maps triangle ABC onto triangle DEF?

(2) a reflection over the y-axis followed by a translation. ABC is flipped over the y-axis, so it is facing the opposite direction. Then it sinks down below the x-axis through a translation.


2.On the set of axes below, the vertices of triangle PQR have coordinates P(-6,7), Q(2,l), and R(-1, -3). (image omitted)
What is the area of triangle PQR?

(3) 25. If you notice, PQ is perpendicular to QR. The slope of PQ is -3/4, and the slope of QR is 4/3. Pythagorean theorem -- as well as knowledge of Pythagorean Triples -- will tell you that PQ = 10 and QR = 5. The area of a triangle is (1/2)(b)(h) = (1/2)(5)(10) = 25.
(You might also remember for several of my comics that the area of a triangle could be written as "one-half a b".)


3.In right triangle ABC, m<C = 90°. If cos B = 5/13, which function also equals 5/13?

(3) sin A. If C is the right angle, then A and B are the acute angles, which are complimentary. The sine of one acute angle will always equal the cosine of the other acute angle, but what is opposite one angle is adjacent to the other.


4.In the diagram below, m<ABC = 268°. (image omitted)
What is the number of degrees in the measure of <ABC?

(4) 46°. The complete circle is 360°. If major arc is ABC = 268, then AC = 360 - 268 = 92. The inscribed angle is half the size of the central angle, so 92 / 2 = 46°.


5.Given triangle MRO shown below, with trapezoid PTRO, MR = 9, MP = 2, and PO = 4. (image omitted)
What is the length of TR?

(4) 6. PO / MO = TR / MR. MO = 6, so 4 / 6 = TR / 9. Cross-multiply and get TR = 6.


6.A line segment is dilated by a scale factor of 2 centered at a point not on the line segment. Which statement regarding the relationship between the given line segment and its image is true?

(3) The line segments are parallel, and the image is twice the length of the given line segment. The image will also be twice as far away from the center than the original line was.


7.Which figure always has exactly four lines of reflection that map the figure onto itself?

(1) Square. Lines through the midpoints of each pair of parallel sides and the two diagonals. An octagon has eight lines: through the four pairs of parallel lines and lines through the opposites vertices.


8.In the diagram below of circle 0, chord D F bisects chord BC at E. (image omitted)
If BC = 12 and FE is 5 more than DE, then FE is

(2) 9. BC is bisected, so BE = CE = 6. The rule for intersecting chords tells us that

(BE)(CE) = (DE)(FE).
(6)(6) = (x)(x + 5)
36 = x2 + 5x

Stop here. If you did this, you're doing too much work for a multiple choice question.

What are the factors of 36?
1, 36; 2, 18; 3, 12; 4, 9; 6, 6.
Which two have a difference of 5? 4 and 9, both of which are choices.
FE is the longer one (it's "5 more"), so the answer is 9.


9.Kelly is completing a proof based on the figure below. (image omitted)
She was given that <A = <EDF, and has already proven AB = DE. Which pair of corresponding parts and triangle congruency method would not prove triangle ABC = triangle DEF?

(2) BC = EF and SAS. SAS requires that the congruent angle be included between the two pairs of congruent sides. BC and EF would give you SSA instead, and that is not allowed (except in the special case of right triangles, where it is HL).


10. In the diagram below, DE divides AB and AC proportionally, m<C = 26°, m<A = 82°, and DF bisects LBDE. (image omitted)
The measure of angle DFB is

(2) 54°. DE divides the two sides proportionally, so DE is parallel to side BC. Angle AED = 26 because it is a corresponding angle. Using the Exterior Angle theorem, we know that <EDB = 82 + 26 = 108. (If you forgot that theorem, you could have found <ADE because a triangle has 180 degrees and then found DF bisects <BDE, so <BDF = <EDF = 54. Angle DFB is an alternate interior angle to EDF, so it equals 54°.

You also could have solved it by finding the angles of triangle DBF. Angle B = 180 - 82 - 26 = 72. So <DFB = 180 - 54 - 72 = 54


11.Which set of statements would describe a parallelogram that can always be classified as a rhombus?
I. Diagonals are perpendicular bisectors of each other.
II. Diagonals bisect the angles from which they are drawn.
III. Diagonals form four congruent isosceles right triangles.

(2) I and III. The diagonals of a rhombus are NOT angle bisectors, unless it's a square.
Note: There is some important information in this question that could have helped you with the proof in Part IV!


12.The equation of a circle is x2 + y2 - 12y + 20 = 0. What are the coordinates of the center and the length of the radius of the circle?

(2) A = 1000(1 + 0.013)2. Interest increases your value, so choices (1) and (3) are right out. The percent 1.3% must be converted to a decimal, which is 0.013. The correct answer is (2). You need to complete the square. Half of -12 is -6, so we need to get to (y - 6)2:

x2 + y2 - 12y + 20 = 0
x2 + y2 - 12y + 36 + 20 = 36
x2 + y2 - 12y + 36 = 16
x2 + (y - 6)2 = 42
This make the center (0, 6) and the radius 4.


13. In the diagram of triangle RST below, m<T = 90°, RS = 65, and ST = 60. (image omitted) What is the measure of <S, to the nearest degree?

(1) 23°. First of all, I hope you realized that RT is the shortest side of the triangle (not because of looks, but because of Pythagorean Triples), so choices (3) and (4) are too big.
We have the adjacent side and the hypotenuse so we need to use cosine to find the angle.
cos S = 60 / 65
S = cos-1 (60 / 65) = 22.62..., which rounds to 23.


14. Triangle A'B'C' is the image of triangle ABC after a dilation followed by a translation. Which statement(s) would always be true with respect to this sequence of transformations?
I. Triangle ABC = triangle A'B'C'
II. Triangle ABC ~ triangle A'B'C'
III. AB || A'B'
IV. AA'= BB'

As a result of discrepancies in the wording, Questions 14 does not have one clear and correct answer.
Either (1) II, only or (3) II and III were accepted.
I don't know what the discrepancy might be that would make someone not think that AB was not parallel to A'B', unless there were printing errors in some books that left out one of the accent marks.

A dilation will preserve the shape of the original object, so the original and the image must be similar.
The center point is not given, but regardless, the orientation will not change, so the slopes of the sides will not change, so the sides will be parallel.
(3) is the best answer, unless some books have a typo of some kind.


15. Line segment RW has endpoints R(-4,5) and W(6,20). Point P is on RW such that RP:PW is 2:3. What are the coordinates of point P?

(2) (0,11). Add 2 + 3 = 5, so RP is 2/5 the length of RW. Find difference of the x-values and y-values of RW and multiply by 2/5.
2/5(6 - (-4)) = 2/5(10) = 4
2/5(20 - 5) = 2/5(15) = 6
Add +4,+6 to point R(-4, 6). P(-4 + 4, 5 + 6) = P(0, 11).


16. The pyramid shown below has a square base, a height of 7, and a volume of 84. (image omitted)
What is the length of the side of the base?

(1) 6. Volume = (1/3)(Area of Base)(height)
84 = (1/3)(B)(7)
252 = 7B
36 = Area of the Base.
The base is a square, then the length of one side is the square root of 36, which is 6


17.In the diagram below of triangle MN 0, LM and LO are bisected by MS and OR, respectively. Segments MS and OR intersect at T, and m<N = 40°. (image omitted)
If m<TMR = 28°, the measure of angle OTS is

(4) 70°. Again, we can use the Exterior Angle Theorem to figure this out.
TMR = 28, but since MS bisects <M, <TMO is also 28, and <OMN = 56°. Since <N = 40°, <MON = 180 - 40 - 56 = 84°. Since OR bisects <MON, then <MOT and <SOT are each 42°.
Look at triangle MOT. If <TMO = 28 and <MOT = 42, then <OTS = 28 + 42 = 70°.


18.In the diagram below, right triangle ABC has legs whose lengths are 4 and 6. (image omitted)
What is the volume of the three-dimensional object formed by continuously rotating the right triangle around AB?

(1) 32 pi. First, rotating the triangle creates a cone. The Volume of a cone is (1/3)(Area of the Base)(height). The area of the base is (pi)(r)2, where the radius is 4. The height is 6 because AB is the axis the triangle is rotated about.
V = (1/3)(pi)(4)2(6) = (1/3)(16)(pi)(6) = 32 pi.

If you look at the incorrect answers: (2) assumes you used 6 as the radius (rotating about AC), (3) assumes you forgot the 1/3, and (4) assumes that you made both mistakes.


19.What is an equation of a line that is perpendicular to the line whose equation is 2y = 3x - 10 and passes through (-6,1)?

(2) y = -2/3 x - 3. The slope of the line in the question is 3/2. The slope of a perpendicular line must be -2/3 because it is the inverse reciprocal, so choices (3) and (4) are out.
Substitute -6 for x in choices (1) and (2).
(1) y = -2/3 (-6) - 5 = 4 - 5 = -1. No good. We don't want (-6, -1)
(2) y = -2/3 (-6) - 3 = 4 - 3 = 1. Check.


20. In quadrilateral BLUE shown below, BE ~ UL. (image omitted)
Which information would be sufficient to prove quadrilateral BLUE is a parallelogram?

(2) LU || BE. If a pair of line segments in a quadrilateral are both parallel and congruent, then the quadrilateral in a parallelogram. This information could be used in the proof in Part IV.
Note that choices (3) and (4) only show that three sides are congruent, not four. That could be a trapezoid.


21. A ladder 20 feet long leans against a building, forming an angle of 71° with the level ground. To the nearest foot, how high up the wall of the building does the ladder touch the building?

(4) 19 You know the length of the ladder, which is the hypotenuse of the right triangle formed by the ladder, the ground and the wall. The wall is opposite the 71° angle. So you need to use sine to find the answer.

So sin 71° = x / 20
x = 20 sin 71° = 18.91 = 19 feet.


22.In the two distinct acute triangles ABC and DEF, <B = <E. Triangles ABC and DEF are congruent when there is a sequence of rigid motions that maps

As a result of discrepancies in the wording, Questions 22 does not have one clear and correct answer. As a result, all choices were accepted.




23.A fabricator is hired to make a 27-foot-long solid metal railing for the stairs at the local library. The railing is modeled by the diagram below. The railing is 2.5 inches high and 2.5 inches wide and is comprised of a rectangular prism and a half-cylinder. (image omitted)
How much metal, to the nearest cubic inch, will the railing contain?

(1) 151. A reminder: there is no simple one-step formula that will give you this answer!
First you have to differentiate how much of the 2.5 inch height is part of the rectangular prism and what is part of the half-cylinder. The diameter of the half-cylinder is 2.5 in, which means that the radius is 1.25 in. Therefore the height of the prism is 1.25.

So the Volume of the railing = the Volume of the prism + the Volume of the half-cylinder
V = L * W * H + (1/2) (pi) (r)2 h
V = (2.5)(1.25)(27) + (1/2)(pi)(1.25)2(27)
V = 84.375 + 66.268..
V = 150.643... = 151




24.In the diagram below, AC = 7.2 and CE = 2.4. (image omitted)
Which statement is not sufficient to prove triangle ABC ~ triangle EDC?

(2) DE = 2.7 and AB = 8.1. In Choice (1), the parallel lines create alternate interior angles that are congruent, so the triangles are similar by AA (or AAA). The other three choices are show ratios that are 3:1, so the numbers themselves are not important. What is important is which sides are given. Keep in mind that there are vertical angles, so we do have one pair of congruent angles. So the other choices need to give us SAS or SSS. Choice (2) gives us SSA, which isn't allowed.

End of Part I

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.

2 comments:

Alan Livingston said...

You can prove AA.

See Benjamin Catalfo's work at http://imgur.com/a/nB3V1

Alan Livingston said...

The previous comment is about Q.24.