If AR = 13 and HR = 12, what is the measure of angle Y, to the nearest degree?

The post-obit are some of the multiple questions from the recent August 2018 New York State Common Core Geometry Regents exam.
The answers to Part Two can exist found here
The answers to Parts 3 and Iv can be found here

Baronial 2018 Geometry, Part I

Each correct answer is worth up to 2 credits. No fractional credit. Work need not be shown.

1. In the diagram beneath, AEFB || CGD, and GE and GF drawn.

If m∠EFG = 32° and k∠AEG 137°, what is chiliad∠EGF?

Answer: (four) 105°
If ∠AEG = 137, then ∠FEG = 180 – 137 = 43.
The sum of the angles of triangle EFG is 180.
And then 180 – (43 + 32) = 105. Bending EGF is 105°.

Alternatively, Angle AEG is an outside bending to triangle EFG. The Remote Angle Theorem says that it must be the sum of the two remote angles EFG and and EGF.
Then EGF = 137 – 32 = 105 degrees.

two. If triangle ABC is mapped onto triangle DEF later a line reflection and triangle DEF is mapped onto triangle XYZ after a translation, the relationship between triangle ABC and triangle XYZ is that they are always

Answer: (ane) congruent and similar
Reflections and translations are rigid motions which preserve the shape of the object. Also, choice (two) coinciding but not similar is not possible. If two triangles are congruent, they are automatically similar.

3. An isosceles right triangle whose legs measure 6 is continuously rotated about one of its legs to form a three-dimensional object. The three-dimensional object is a

Answer: (4) cone with a bore of 12
When a right triangle is rotated about a leg, the resulting 3-D shape will be a cone. Considering the base of the triangle is half dozen, the radius will be 6, and so the diameter is twice that, or 12.

four. In regular hexagon ABCDEF shown below, Advertisement, BE, and CF all intersect at G.

When triangle ABG is reflected over BG and and then rotated 180° about indicate M, triangle ABG is mapped onto

Answer: (one) Triangle FEG
When it is reflected, it is mapped onto triangle BGC, the top center of the hexagon. When information technology is rotated 180 degrees, information technology maps onto FEG, the bottom center of the hexagon.

5. A right cylinder is cut perpendicular to its base. The shape of the cross section is a

Answer: (3) rectangle
A horizontal (parallel to base) cut would give a circle. A vertical (perpendicular to base of operations) cutting gives a rectangle. If you look at a tin can on a shelf, information technology appears to be a rectangle.

vi. Yolanda is making a springboard to use for gymnastics. She has 8-inch-alpine springs and wants to grade a 16.5° angle with the base, as modeled in the diagram below.

To the nearest tenth of an inch, what will be the length of the springboard, ten?

Answer: (4) 28.2
Opposite and hypotenuse hateful you need to use sine.
sin 16.5 = 8 / ten
x = eight / sin xvi.5 = 28.167…

7. In the diagram beneath of right triangle ABC, altitude BD is drawn to hypotenuse Air-conditioning.

If BD = 4, AD = x – half dozen, and CD = x what is the length of CD?

Answer: (3) 8
According to the Right Triangle Altitude Theorem, (BD)² = (Advertisement)(CD)
So 4² = (x – six)(x)
16 = 10² - 6x. At this point, yous could substitute the choices, or solve.
x² - 6x – 16 = 0
(10 – 8)(x + ii) = 0
x = eight or x = -2. Discard the negative length.

viii. Rhombus STAR has vertices Southward(–1,2), T(2,three), A(3,0), and R(0,–1). What is the perimeter of rhombus STAR?

Answer: (4) 4 * SQRT(x)
Each side of the rhomb has the aforementioned length, so calculate the length of one side using the distance formula, and multiply the upshot past 4.
SQRT( (two - -ane)² + (iii-2)ⁱⁱ ) = SQRT( (3)² + (1)² )
= SQRT(nine + i) = SQRT(10) per side.
Perimeter is 4 * SQRT(10).

9. In the diagram beneath of triangle HAR and triangle NTY, angles H and N are right angles, and HAR ~ NTY.

If AR = 13 and HR = 12, what is the measure out of bending Y, to the nearest degree?

Answer: (i) 23°
Angle Y corresponds to angle R. AR is the hypotenuse, and Hour is next to R. This means use cosine to find the size of bending R, which volition also be the size of angle Y.
cos R = 12/xiii
r = cos^-1(12/xiii) = 22.61 = 23 degrees. Note that choice (4) is the angle if y'all either used sine, or if yous solved for bending A. You would get choices (ii) and (iii) if yous used tangent, and if you lot used tangent and the incorrect angle.

10. In the diagram below, AKS, NKC, AN, and SC are drawn such that AN = SC.

Which additional argument is sufficient to evidence triangle KAN = triangle KSC by AAS?

Answer: (4) AN || SC
If AN is parallel to SC, and so you would become alternate interior angles, along with the vertical angles. That is plenty to prove congruency by either ASA or AAS. Pick (1) would prove congruency by SSS, merely wouldn't requite any more information virtually the angles. Choice (2) would yield SSA, which is not sufficient to prove congruency. Choice (3) would institute that the vertical angles are as well right angles, just we already know that those ii angles are congruent.

11. Which equation represents a line that is perpendicular to the line represented by y = (2/iii)x + 1 ?

Answer: (1) 3x + 2y = 12
The negative reciprocal of 2/3 is -3/2, and then choices (3) and (four) are both incorrect.
y = -(3/2)10 + b
3/2x + y = b, multiply past two to get rid of the fraction
3x + 2y = 2b, which gives us the starting time equation, if 2b is replaced by 12.

12. In the diagram of ABC below, points D and Due east are on sides AB and CB respectively, such that DE || Air conditioning.

If EB is 3 more DB, AB = 14, and CD = 21, what is the length of AD?

Answer: (two) 8
Be careful because you need to find the length of DB first, merely they are request for the length of Advertising, then y'all accept to subtract DB from AB.
The sides are proportional, so you lot can set an equation:
x / 14 = (ten + iii) / 21. You can substitute the choices from this point if you want.
21x = 14x + 42
7x = 42
x = six
AD = 14 – 6 = viii.
Notice that the four choices represent the lengths of DB, AD, Be, and CE. You might have reasoned this one out from the diagram and then checked your work to run into if y'all were right.

13. Quadrilateral MATH has both pairs of reverse sides congruent and parallel. Which statement about quadrilateral MATH is always true?

Reply: (iv) ∠MAT = ∠MHT
Check Part IV of this exam for a coordinate geometry trouble involving a quadrilateral with vertices Thousand, A, T, and H.
Contrary sides congruent and parallel mean that this is a parallelogram. That means that the opposite angles are also congruent.
Option (one) says that the diagonals must be congruent, which is only true in rectangles. (You could use this in Office Four)
Choice (2) says that the consecutive angles are right angles, which is merely true in rectangles. (Again, Function IV)
Choice (iii) would occur in rectangles as well because the two diagonals would create 4 isosceles triangles.

fourteen. In the effigy shown beneath, quadrilateral TAEO is circumscribed around circle D. The midpoint of TA is R, and HO = PE .

If AP = 10 and EO = 12, what is the perimeter of quadrilateral TAEO?

Answer: (2) 64

AP = ten so AR =10. R is the midpoint of AT, so RT = 10, which means TH = ten.
HO = PE ways that OZ = ZE, and since OE = 12, and then HO = PE = OZ = PE = half-dozen.
four * 10 + four * 6 = 64

15. The coordinates of the endpoints of directed line segment ABC are A(-8,7) and C(7,-xiii). If AB:BC = iii:ii, the coordinates of B are

Answer: (1) (1, -5)
B is 3/5 of the distance from A to C. The x-coordinates of A and C are seven – (-8) = xv units apart.
3/v(15) = ix, and -8 + 9 = ane, so the 10-coordinate of B is 1, which is choice (i).
To check, -13 – vii = -xx, and (3/5)(-xx) = -12, and 7 – 12 = -five, which is the y-coordinate of B.

16. In triangle ABC, points D and E are on sides AB and BC, respectively, such that DE || AC, and AD:DB 3:5.

If DB = 6.3 and Ac = 9.iv, what is the length of DE, to the nearest tenth?

Reply: (3) 5.9

If Advertising:DB = 3:v, then AB:DB = eight:v, which is the ratio of the larger triangle to the smaller triangle.
Then viii/5 = nine.4/x
8x = (5)(9.4)
x = (5)(9.4)/eight = v.875.

17. In the diagram below, rectangle ABCD has vertices whose coordinates are A(vii,1), B(nine,3), C(three,9), and D(1,7).

Which transformation will not comport the rectangle onto itself?

Answer: (iii) a rotation of 180° nearly the point (6,6)
The rectangle will carry onto itself in a rotation well-nigh the rectangle'due south center. (5, 5) is the eye of the rectangle. (vi, 6) is not. If you lot rotate nearly (half-dozen, vi), the image will be adjacent to the original rectangle.

18. A circle with a bore of 10 cm and a central angle of thirty° is drawn below.

What is the area, to the nearest 10th of a square centimeter, of the sector formed by the 30° angle?

Answer: (2) 6.five
The radius is 5 cm. The cardinal angle is xxx degrees, which is 1/12 of the total circumvolve. (thirty/360 = 1/12)
V = (i/12)(pi)(v)ⁱⁱ = 6.54..

19. A kid's tent tin can exist modeled every bit a pyramid with a square base whose sides measure 60 inches and whose height measures 84 inches. What is the volume of the tent, to the nearest cubic foot?

Respond: (2) 58
60 inches = 5 feet, 84 inches = seven feet
Volume = (ane/3) (Area of the base) (height)
V = (i/3) (5) (5) (7) = 58.333...

20. In the accompanying diagram of right triangle ABC, distance BD is drawn to hypotenuse AC.

Which statement must always exist truthful?

Reply: (2) Advertising/AB = AB/Air-conditioning
Brusque leg is to hypotenuse equally short leg is to hypotenuse.
The other choices exercise non apply corresponding sides in the correct society.

21. An equation of circumvolve O is x² + y² + 4x - 8y = -16. The statement that all-time describes circumvolve O is the

Answer: (four) centre is (-two,iv) and is tangent to the y-centrality
Rewrite the equation in standard course past grouping the variables and completing the foursquare.
x² + y² + 4x - 8y = -sixteen
ten² + 4x + yⁱⁱ - 8y = -16
10² + 4x + iv + y^two - 8y + sixteen = -16 + 4 + sixteen
(ten + two)² + (y - iv) = 4 = ii²
The center of the circle is (-2, 4) and the radius is 2, which would make it tangent to the y-axis.

22. In triangle ABC, BD is the perpendicular bisector of ADC. Based upon this information, which statements below tin be proven?

I. is a median.
2. bisects ∠ABC.
Three. ABC is isosceles.

Answer: (4) I, Two, and III
If BD is a bisector of ADC, information technology'southward also a median.
If BD is a perpendicular bisector, than AB and Ac must exist congruent. (This comes in handy to know in Part Ii!)
IF AB = AC, the triangle is isosceles. In an isosceles triangle, the median bisects the vertex angle.

23. Triangle RJM has an area of 6 and a perimeter of 12. If the triangle is dilated by a scale cistron of 3 centered at the origin, what are the expanse and perimeter of its paradigm, triangle R'J'Thou'?

Answer: (3) surface area of 54 and perimeter of 36
If the image is dilated past a scale cistron of three, then the perimeter is increased by a gene of iii, just the area is increased past a gene of three² = ix.
12 * iii = 36, and 6 * 9 = 54.

24. If sin (2x + vii)° = cos (4x - seven)°, what is the value of x?

Answer: (2) 15
The Sine of whatever bending is equal to the Cosine of the complementary angle. And the sum of the two complementary angles is 90 degrees
2x + 7 + 4x - 7 = ninety
6x = 90
x = 15

Cease of Part I

How did yous do?

Questions, comments and corrections welcome.

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Source: http://mrburkemath.blogspot.com/2019/05/august-2018-common-core-geometry_9.html

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