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### June 2017, Geomeattempt (Common Core), Part I

**1.**

*In the diagram below, triangle ABC = triangle DEF. (picture omitted)Which sequence of revolutions 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 dvery own below the x-axis via a translation.

**2.**

*On the set of axes below, the vertices of triangle PQR have coordinates P(-6,7), Q(2,l), and also R(-1, -3). (photo omitted)What is the location of triangle PQR?*(3) 25. If you notice, PQ is perpendicular to QR. The slope of PQ is -3/4, and also the slope of QR is 4/3. Pythagorean theorem -- and also expertise of Pythagorean Triples -- will certainly tell you that PQ = 10 and also QR = 5. The area of a triangle is (1/2)(b)(h) = (1/2)(5)(10) = 25.(You might also remember for a number of of my comics that the area of a triangle could be created as "one-fifty percent a b".)

**3.**

*In appropriate triangle ABC, m<C = 90°. If cos B = 5/13, which feature also amounts to 5/13?*(3) sin A. If C is the best angle, then A and also B are the acute angles, which are free. The sine of one acute angle will constantly equal the cosine of the other acute angle, however what is opposite one angle is adjacent to the other.

**4.**

*In the diagram listed below, m<ABC = 268°. (picture omitted)What is the variety of degrees in the meacertain of <ABC?*(4) 46°. The finish circle is 360°. If major arc is ABC = 268, then AC = 360 - 268 = 92. The inscribed angle is fifty percent the size of the central angle, so 92 / 2 = 46°.

**5.**

*Given triangle MRO shown listed below, via trapezoid PTRO, MR = 9, MP = 2, and PO = 4. (picture omitted)What is the size of TR?*(4) 6. PO / MO = TR / MR. MO = 6, so 4 / 6 = TR / 9. Cross-multiply and also acquire TR = 6.

**6.**

*A line segment is dilated by a scale variable of 2 focused at a allude not on the line segment. Which statement concerning the relationship in between the provided line segment and also its image is true?*(3) The line segments are parallel, and the image is twice the size of the provided line segment. The image will certainly likewise be twice as much amethod from the center than the original line was.

**7.**

*Which figure always has actually exactly 4 lines of reflection that map the number onto itself?*(1) Square. Lines through the midpoints of each pair of parallel sides and also the two diagonals. An octagon has eight lines: through the four pairs of parallel lines and lines with the opposites vertices.

**8.**

*In the diagram below of circle 0, chord D F bisects chord BC at E. (picture omitted)If BC = 12 and also FE is 5 more than DE, then FE is*(2) 9. BC is bisected, so BE = CE = 6. The preeminence for intersecting chords tells us that (BE)(CE) = (DE)(FE). (6)(6) = (x)(x + 5)36 = x2 + 5xStop right here. If you did this, you"re doing as well a lot work-related for a multiple alternative question.What are the factors of 36?1, 36; 2, 18; 3, 12; 4, 9; 6, 6.Which 2 have a difference of 5? 4 and 9, both of which are selections.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. (photo omitted)She was given that <A = <EDF, and also has actually currently prstove AB = DE.Which pair of equivalent components and also triangle congruency methodwould certainly not prove triangle ABC = triangle DEF?*(2) BC = EF and also SAS. SAS needs that the congruent angle be contained in between the two pairs of congruent sides. BC and also EF would certainly provide you SSA rather, and also that is not enabled (other than in the distinct case of right triangles, wright here it is HL).

**10.**

*In the diagram listed below, DE divides AB and also AC proportionally,m<C = 26°, m<A = 82°, and DF bisects LBDE. (picture omitted)The measure of angle DFB is*(2) 54°.DE divides the 2 sides proportionally, so DE is parallel to side BC. Angle AED = 26 because it is a equivalent angle. Using the Exterior Angle theorem, we know that <EDB = 82 + 26 = 108. (If you forobtained that theorem, you could have discovered <ADE because a triangle has actually 180 levels and then discovered <EDB bereason it is supplementary.)DF bisects <BDE, so <BDF = <EDF = 54. Angle DFB is an alternative interior angle to EDF, so it equals 54°.You also could have resolved it by finding the angles of triangle DBF. Angle B = 180 - 82 - 26 = 72. So <DFB = 180 - 54 - 72 = 54

**11.**

*Which collection of statements would certainly explain a parallelogram that canalways be classified as a rhombus?I. Diagonals are perpendicular bisectors of each various other.II. Diagonals bisect the angles from which they are attracted.III. Diagonals create four congruent isosceles ideal triangles.*

**Update:**

**(4) I, II and III. The diagonals of a rhombus will divide it right into 4 congruent isosceles triangles. This deserve to only take place if they are perpendicular to each other, bisect each other and also bisect the angles.Note:**Tright here is some crucial indevelopment in this question that could have helped you through the proof in Part IV!(2) I and III. The diagonals of a rhombus are NOT angle bisectors, unless it"s a square.Note: Tright here is some essential indevelopment in this question that might have actually aided you via the proof in Part IV!I don"t recognize wright here my head was once I typed the above. Obviously, it"s a rectangle wright here the diagonals only bisect the angles as soon as it is also a square. Just reflects you that you have to rush yourself, and also to always double check. If the angles weren"t bisected, how might III by true? Looks like I would certainly not have actually scored 100 on this test.

**12.**

*The equation of a circle is x2 + y2 - 12y + 20 = 0. What are thecoordinates of the center and also the length of the radius of the circle?*(2) A = 1000(1 + 0.013)2. Interest increases your worth, so choices (1) and also (3) are best out. The percent 1.3% need to 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 should obtain to (y - 6)2: x2 + y2 - 12y + 20 = 0x2 + y2 - 12y + 36 + 20 = 36x2 + y2 - 12y + 36 = 16x2 + (y - 6)2 = 42This make the facility (0, 6) and also the radius 4.

**13.**

*In the diagram of triangle RST listed below, m<T = 90°, RS = 65, and ST = 60. (image omitted)What is the meacertain of <S, to the nearemainder degree?*(1) 23°. First of all, I hope you realized that RT is the shortest side of the triangle (not because of looks, yet bereason of Pythagorean Triples), so options (3) and (4) are also huge.We have the adjacent side and the hypotenuse so we have to usage cosine to uncover the angle.cos S = 60 / 65S = cos-1 (60 / 65) = 22.62..., which rounds to 23.

**14.**

*Triangle A"B"C" is the picture of triangle ABC after a dilation adhered to by a translation.Which statement(s) would constantly be true via 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 an outcome of inequalities in the wording, Questions 14 does not have actually one clear and also correct answer.**Either (1) II, just or (3) II and III were accepted.I do not recognize what the discrepancy might be that would certainly make someone not think that AB was not parallel to A"B", unmuch less there were printing errors in some books that left out one of the accent marks.A dilation will certainly preserve the form of the original object, so the original and also the image must be equivalent.The center point is not offered, but regardmuch less, the orientation will certainly not adjust, so the slopes of the sides will not readjust, so the sides will certainly be parallel.Update: (3) is an acceptable answer bereason it is possible, considering that specific worths were not given, that after the translation, the dilation could reason AB and A"B" to become component of the same line, and also not parallel. (Thank you to those who commented and emailed me.)

**15.**

*Line segment RW has actually endpoints R(-4,5) and W(6,20). Point P ison RW such that RP:PW is 2:3. What are the collaborates 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-worths of RW and also multiply by 2/5.2/5(6 - (-4)) = 2/5(10) = 42/5(20 - 5) = 2/5(15) = 6Add +4,+6 to point R(-4, 6). P(-4 + 4, 5 + 6) = P(0, 11).

**16.**

*The pyramid displayed listed below has a square base, a height of 7, and also a volume of 84. (picture omitted)What is the size of the side of the base?*(1) 6. Volume = (1/3)(Area of Base)(height)84 = (1/3)(B)(7)252 = 7B36 = Area of the Base.The base is a square, then the size of one side is the square root of 36, which is 6

**17.**

*In the diagram listed below of triangle MNO, <M and also <O are bisected by MS and OR, respectively. Segments MS and also OR intersect at T, and m<N = 40°. (picture omitted)If m<TMR = 28°, the meacertain of angle OTS is*(4) 70°. Aacquire, we deserve to usage the Exterior Angle Theorem to figure this out.TMR = 28, yet considering that MS bisects <M, <TMO is likewise 28, and also <OMN = 56°. Because <N = 40°, <MON = 180 - 40 - 56 = 84°. Due to the fact that OR bisects <MON, then <MOT and also <SOT are each 42°.Look at triangle MOT. If <TMO = 28 and also <MOT = 42, then <OTS = 28 + 42 = 70°.

**18.**

*In the diagram listed below, appropriate triangle ABC has actually legs whose lengths are 4 and also 6. (photo omitted)What is the volume of the three-dimensional object developed by repetitively rotating the right triangle roughly 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, wbelow the radius is 4. The elevation is 6 bereason 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 provided 6 as the radius (rotating around 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 whoseequation is 2y = 3x - 10 and also 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 need to be -2/3 bereason it is the inverse reciprocal, so choices (3) and (4) are out.Substitute -6 for x in options (1) and also (2).(1) y = -2/3 (-6) - 5 = 4 - 5 = -1. No good. We do not want (-6, -1)(2) y = -2/3 (-6) - 3 = 4 - 3 = 1. Check.

**20.**

*In quadrilateral BLUE presented listed below, BE ~ UL. (image omitted)Which indevelopment would be adequate 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 selections (3) and also (4) only show that three sides are congruent, not 4. That could be a trapezoid.

**21.**

*A ladder 20 feet lengthy leans versus a structure, creating an angle of 71° with the level ground. To the nearemainder foot, how high up the wevery one of the structure does the ladder touch the building?*(4) 19 You recognize the length of the ladder, which is the hypotenuse of the best triangle created by the ladder, the ground and also the wall. The wall is oppowebsite the 71° angle. So you should use sine to find the answer.So sin 71° = x / 20x = 20 sin 71° = 18.91 = 19 feet.

**22.**

*In the two distinct acute triangles ABC and also DEF, <B = <E.Triangles ABC and also DEF are congruent as soon as tbelow is a sequence ofrigid movements that maps*As an outcome of imbalances in the wording, Questions 22 does not have actually one clear and also correct answer.As a result, all options were welcomed.

**23.**

*A fabricator is hired to make a 27-foot-long solid steel railing forthe stairs at the neighborhood library. The railing is modeled by the diagrambelow. The railing is 2.5 inches high and also 2.5 inches wide and also iscomprised of a rectangular prism and a half-cylinder. (photo omitted)How much steel, to the nearemainder cubic inch, will the railing contain?*(1) 151. A reminder: there is no straightforward one-step formula that will offer you this answer!First you have to distinguish exactly how a lot of the 2.5 inch elevation is component of the rectangular prism and also what is part of the half-cylinder.The diameter of the half-cylinder is 2.5 in, which suggests that the radius is 1.25 in. As such the height of the prism is 1.25.So the Volume of the railing = the Volume of the prism + the Volume of the half-cylinderV = L * W * H + (1/2) (pi) (r)2 hV = (2.5)(1.25)(27) + (1/2)(pi)(1.25)2(27)V = 84.375 + 66.268..V = 150.643... = 151CORRECTION: I usually look for such things, yet I skipped over converting the feet into inches. That"s what I gain for not checking my occupational versus the answer key when it came to be available! (And many thanks for pointing this out.)(3) 1808. A reminder: there is no easy one-step formula that will provide you this answer!First you have to differentiate how much of the 2.5 inch elevation is component of the rectangular prism and what is part of the half-cylinder.The diameter of the half-cylinder is 2.5 in, which indicates that the radius is 1.25 in. Because of this the height of the prism is 1.25.So the Volume of the railing = the Volume of the prism + the Volume of the half-cylinderV = L * W * H + (1/2) (pi) (r)2 h(27) * (12) = 324V = (2.5)(1.25)(324) + (1/2)(pi)(1.25)2(324)V = 1012.5 + 795.215..V = 1807.7 = 1808

**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 produce alternative internal angles that are congruent, so the triangles are equivalent by AA (or AAA). The other three selections are show ratios that are 3:1, so the numbers themselves are not important. What is necessary is which sides are provided. Keep in mind that tright here are vertical angles, so we carry out have one pair of congruent angles. So the various other options need to give us SAS or SSS. Choice (2) offers us SSA, which isn"t permitted.

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**Update:**

**As listed in the comments below, this question, as written, has actually NO**answer. It has a "best" answer, which can be said only indicates that it"s the "leastern wrong", which is still wrong. The Regents Board has proclaimed that they are not rescoring the exams. Actually, I do not blame them -- the test is poor sufficient without making it worse. Allow me to explain my allude of view:Two various other inquiries on this test had troubles. One was invalidated bereason tbelow was no positive correct answer, so everyone received credit. In one more case, they welcomed 2 answers for which crmodify was provided. The passing grade for this exam is currently artificially low. I would like a much less confutilizing exam (I wouldn"t say "much less rigorous", however possibly that, too) with a higher threshold for passing, and a curve which does not punish the high-achievers.I will not try to suggest that the Regents exam models real life any type of even more than the SATs do. But test taking itself is a ability, and also sometimes you execute need to uncover the "best" solution, or even the "leastern worst", and not the "right" or the "one true" answer.Now, I can go either means, relying on the question. If the wording is so ambiguous regarding leave a student guessing between two various options depending upon exactly how you read something, that"s a no-brainer: toss it. And if something is left out so that tright here is no positive answer, that have to be instantly thrvery own out as well.Inquiry 24, on the other hand also, is a little various. First of all, you aren"t in search of a positive answer at all; you want an unfavorable answer. Which one is NOT enough, which indicates that you must have the ability to display that at least three of them are. Keeping in mind that this is a multiple-option question, students are not intended to display any kind of work-related at all, though they are motivated to do so in the margins of the book. They are NOT meant to compose two-web page proofs in the margins, and most would certainly not.Now, let me sheight and also say this: I applaud any kind of student that has the determicountry to compose a two-web page proof in the margins.If you were to answer this Geomeattempt question around similarity, the initially action would certainly be use the Laws of Similarity and also Congruence. These are quick to usage and also easily eliminate three of the four selections. At this allude, one might pick the fourth alternative (Choice 2) as being the correct response. At the very leastern, it is the one the Regents were searching for. (Yes, I hate when teachers make you guess the solution that they are in search of, whether or not it is the one that makes the many feeling -- However before, in this case, it is the one that provides the many feeling.)I was troubled by this question when I initially observed it, because it provided specific numbers. I originally thought it would certainly be a calculation trouble, but the numbers confirmed little usage in the techniques presented over. That doesn"t mean that they couldn"t be helpful for various techniques. In particular, you can usage the Law of Sines to compare the ratios and find the angle in between the sides. Given that there is a pair of vertical angles in the diagram, that is sufficient to prove similarity by AA. Instead of a logic-based argument, you can carry out a calculation-based discussion.But here"s the thing: asking a student to carry out that in an Open-Ended trouble would certainly be difficult enough that I wouldn"t also expect to uncover it in Part II. That would certainly be a Part III question, at leastern, worth a minimum of 4 points. No one would mean any kind of student, especially one that "reperceived for the test" -- sadly, a important ability these days -- to perform any of that on a multiple-choice question. Again, to those of you that go these added lengths: More Power to You! But, sadly, utilizing the Law of Sines in this situation probably wouldn"t take place to a lot of students also if, a) they were taught the idea properly, and also b) they could execute it to its conclusion. Finally, this isn"t a situation of confusion, but one of omission. Tright here actually is a quick deal with for this problem. Had the question declared "with the rules for Similarity and also Congruency" (or something similar), their answer would be valid, but I suspect that such an enhancement would be considered as providing too much away.Oddly, it likewise could have actually been corrected had actually they switched the numbers about so that AB and DE were not the longest sides, which is an essential part of Benjamin Catalfo"s proof, yet I doubt that they intended anyone to go this far on a multiple-choice question. Hopefully, they are currently hence informed and won"t make the same mistake twice. Likewise, I hope that Mr. Catalfo, upon finding no correct answer, was wise enough to choose the one that they were searching for. Sadly, occasionally this is how math -- and education and learning, in general -- models the "actual world".End of Part IHow did you do? Comments, inquiries, corrections and also involves are all welcome. Typos happen.

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