## Tuesday, February 27, 2018

### January 2018 Common Core Geometry Regents, Part 1 (mult choice)

The following are some of the multiple questions from the recent January 2018 New York State Geometry Regents exam.

### January 2018 Geometry, Part I

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

1. In the diagram below, a sequence of rigid motions maps ABCD onto JKLM.

If m<A = 82°, m<B = 104°, and m<L = 121°, the measure of <M is

Because they were rigid motions, the corresponding angles are congruent. A corresponds to J, which is 82 degrees; B corresponds to K, which is 104 degrees. L is given as 121 degrees. The quadrilateral has a total of 360 degrees.
360 - (82 + 104 + 121) = 53

2. Parallelogram HAND is drawn below with diagonals HN and AD intersecting at S.

Which statement is always true?

Answer: (2) AS = 1/2 AD .
The diagonals of a parallelogram bisect each other.

3. The graph below shows two congruent triangles, ABC and A'B'C'.

Which rigid motion would map triangle ABC onto triangle A'B'C'?

Answer: (4) a reflection over the line y = x.
A translation wouldn't change the orientation. If there were a rotation, each point would have more 1 quadrant (for 90 degrees) or 2 quadrants (for 180 degrees).

4. A man was parasailing above a lake at an angle of elevation of 32° from a boat, as modeled in the diagram below.

If 129.5 meters of cable connected the boat to the parasail, approximately how many meters above the lake was the man?

The height is opposite the 32° angle, and the hypotenuse of 129.5 is given. Use the sine ration: sin = opp/hyp
sin 32 = x / 129.5
x = 129.5 (sin 32) = 68.6245447...
Be sure to have the calculator in DEGREE mode! If the calculator is set to radians, you will get an incorrect answer that might seem to be reasonable (71.4), but isn't one of the choices.
Speaking of "reasonable", because 32 degrees is close to 30 degrees, you could have estimated a multiple choice answer as follows: in a 30-60-90 degree triangle, the side opposite the 30 degree angle will ALWAYS be half of the hypotenuse. In that case the height would be 64.75. Since the angle is 32 degrees, it should be a little bigger than this number. Only 68.6 would be reasonable.

5. A right hexagonal prism is shown below. A two-dimensional cross section that is perpendicular to the base is taken from the prism.

Which figure describes the two-dimensional cross section?

Parallel to the base would give a hexagon. Perpendicular to the base, whether it's "front to back", "side to side" or any other direction will yield a rectangle.

6. In the diagram below, AC has endpoints with coordinates A(-5,2) and C(4, -10).

If B is a point on AC and AB:BC = 1:2, what are the coordinates of B?

A ratio of 1:2 means that AB is 1/3 the length and BC is 2/3 the length.
If you look at the change in the x-coordinates, the distance from -5 to 4 is 9. (4 - (-5) = 9.)
One third of 9 is 3, so the change in the x-coordinate from A to B is +3.
-5 + 3 = -2, so the only choice is (-2, 2).
Checking the y-coordinate: (-10) - 2 = -12. (1/3)(-12) = -4.
2 + (-4) = -2, which is the y-coordinate of B.

7. An ice cream waffle cone can be modeled by a right circular cone with a base diameter of 6.6 centimeters and a volume of 54.45(pi) cubic centimeters. What is the number of centimeters in the height of the waffle cone?

V = (1/3)(pi)r2h
54.45(pi) = (1/3)(pi)(3.3)2h
h = 54.45(pi) / ((1/3)(pi)(3.3)2) = 15
You could solve the equation for h first, or you can plug in the numbers first and then solve. Note that you can divide both sides of the equation by pi to remove it from the equation entirely. Note that you were given a diameter of 6.6, which makes the radius 3.3.

8. The vertices of triangle PQR have coordinates P(2,3), Q(3,8), and R(7,3). Under which transformation of triangle PQR are distance and angle measure preserved?

Answer: (4) (x,y) -> (x + 2, y + 3).
Distance is not preserved if one of the coordinates is multiplied, which eliminates choices (1), (2), and (3). Moreover, if the coordinates are not multiplied by the same scale factor, then angle measure will not be preserved either.
Choice (4) is a translation, which preserves distance and angle measure.

9. In triangle ABC shown below, side AC is extended to point D with m<DAB = (180 - 3x)°, m<B = (6x - 40)°, and m<C = (x + 20)°.

What is m<BAC?

According to the Exterior Angle Theorem, the sum of the exterior angle equals the sum of the two remote angles.
Therefore,

(180 - 3x)= (6x - 40) + (x + 20)
180 + 40 - 20 = 6x + x + 3x
200 = 10x
x = 20

The measure of angle BAC is supplementary to DAB.
Notice that 180 - (180 - 3x) = 3x, and 3(20) = 60 degrees.
The longer way: m<DAB = 180 - 3(20) = 180 - 60 = 120, and then M&BAC = 180 - 120 = 60 degrees

10. Circle O is centered at the origin. In the diagram below, a quarter of circle O is graphed.
Which three-dimensional figure is generated when the quarter circle is continuously rotated about the y-axis?

If you reflect the quarter circle, you would get a semicircle in two dimensions. In three dimensions, that would be a hemisphere.

11. Rectangle A'B'C'D' is the image of rectangle ABCD after a dilation centered at point A by a scale factor of 2/3. Which statement is correct?

Answer: (1) Rectangle A'B'C'D' has a perimeter that is 2/3 the perimeter of rectangle ABCD.
The perimeter would shrink. It would only be 2/3 of the original.
The area would be reduced to (2/3)2, or 4/9, of the original area.

12. The equation of a circle is x2 + y2 - 6x + 2y = 6. What are the coordinates of the center and the length of the radius of the circle? (1) center (-3,1) and radius 4 (2) center (3, -1) and radius 4 (3) center (-3,1) and radius 16 (4) center (3, -1) and radius 16

Answer: (2) center (3, -1) and radius 4.
Rewrite the formula into standard form,
(x - h)2 + (y - k)2 = r2, by grouping the variables and then completing the squares:

x2 + y2 - 6x + 2y = 6
x2 - 6x + y2 + 2y = 6
Half of -6 is -3, and (-3)2 = 9. Add 9 to both sides.
x2 - 6x + 9 + y2 + 2y = 6 + 9
Half of 2 is 1, and (1)2 = 1. Add 1 to both sides.
x2 - 6x + 9 + y2 + 2y + 1= 6 + 9 + 1
Factor the polynomials into binomials
(x - 3)2 + (y + 1)2 = 16

The center is (3, -1) and the radius is 4.
Remember to flip the signs to get the coordinates, and take the square root of r2.

13. In the diagram of triangle ABC below, DE is parallel to AB, CD = 15, AD= 9, and AB= 40.

The length of DE is

CD / DE = CA / AB
15 / x = (15 + 9) / 40
(15 + 9)x = (15)(40)
24x = 600
x = 25

14. The line whose equation is 3x - 5y = 4 is dilated by a scale factor of 5/3 centered at the origin. Which statement is correct?

Answer: (1) The image of the line has the same slope as the pre-image but a different y-intercept.
When dilating a line, the slope will not change. The new line will either be parallel to the original line, or coincident (i.e., the same line) if the center of the dilation is a point on the line.
If the center of the dilation is a point on the line, then the y-intercept would not change either, but that is not the case in this example. If the origin, (0, 0), were a point on the line, then 3(0) - 5(0) = 4, which is not true.

15. Which transformation would not carry a square onto itself?

Answer: (3) a 180° rotation about one of its vertices.
A 180° rotation about the center of the square would carry it onto itself.

16. In circle M below, diameter AC, chords AB and BC, and radius MB are drawn.

Which statement is not true?

Answer: (4) mAB = (1/2)m<ACB .
The measure of arc AB would be TWICE the measure of angle ACB. In other words, the measure of angle ACB would be half of arc AB.
ABC must be a right triangle, because angle B must be a right angle because it intercepts a semicircle.
ABM must be isosceles because both AM and BM are radii.
The measure of arc BC is equal to the measure of its central angle, angle BMC.

17. In the diagram below, XS and YR intersect at Z. Segments XY and RS are drawn perpendicular to YR to form triangles XYZ and SRZ.

Which statement is always true?

Answer: (4) XY / SR = YZ / RZ.
We have enough information to show that the two triangles are similar, but not congruent. Eliminate choices (2) and (3).
Angles Y and R are right angles because they are perpendicular to YR, so they are congruent.
Angles XZY and RZS are congruent because they are vertical angles.
Therefore, XYZ is similar to RSZ by AA.
That means that the corresponding sides are proportional in length. Choice (4) is a correct proportion

18. As shown in the diagram below, ABC || EFG and BF = EF.

If m<CBF = 42.5°, then m<EBF is

Because BF = EF, then BEF is an isosceles triangle.
So angle BEF is congruent to angle FBE because they are the base angles and BFE is the vertex angle.
Angle BFE is congruent to angle CBF because they are alternate interior angles.
Then CBF = BFE = 42.5 degrees.
So x + x + 42.5 = 180
2x = 137.5
x = 68.75°

19. A parallelogram must be a rhombus if its diagonals

Answer: (4) are perpendicular to each other .
Diagonals of a rectangle are also congruent, so choice (1) is incorrect.
Diagonals of a rectangle also bisect each other, so choice (2) is incorrect.
Diagonals of a rhombus do bisect each other, so choice (3) is incorrect.

20. What is an equation of a line which passes through (6,9) and is perpendicular to the line whose equation is 4x - 6y = 15?

Answer: (1) y - 9 = (-3/2)(x - 6).
The original line is in Standard Form. Find the slope of the line, using -A/B, or by rewriting it into point-slope or slope-intercept form.
A = 4, B = -6, so the slope is (-4)/(-6) = 2/3
The slope of a line perpendicular to this is -3/2, the negative reciprocal. This eliminates choices (2) and (4).
Point-slope form is y - y0 = m(x - x0), so the solution is y - 9 = (-3/2)(x - 6)

21. Quadrilateral ABCD is inscribed in circle 0, as shown below.
If m<A = 80°, m<B = 75°, m<C = (y + 30)°, and m<D = (x -10)°, which statement is true?

Answer: (4) x = 115 and y = 70.
The sum of angles A and C is 180 degrees, and the sum of angles B and D is 180 degrees.
Angle C is 180 - 80 = 100 degrees, so y = 70.
Angle D is 180 - 75 = 105 degrees, do x = 115.

22. A regular pyramid has a square base. The perimeter of the base is 36 inches and the height of the pyramid is 15 inches. What is the volume of the pyramid in cubic inches?

The perimeter of the square base is 36, so each side of the square is 9 (not 6). The area of the base is 92 = 81.
The volume of the pyramid is (1/3) times the Area of the Base times the height:
V = (1/3)(81)(15) = 405

23. In the diagram below of triangle ABC, <ABC is a right angle, AC = 12, AD = 8, and altitude BD is drawn.
What is the length of BC?

Answer: (2) 4 * SQRT(3).
Triangle BCD is similar to ABC because they both have right angles and they both have angle C (Reflexive Property).
So you can compare leg / hypotenuse = leg / hypotenuse
DC / BC = BC / AC
(12 - 8) / BC = BC / 12
BC2 = (12)(4) = 48
BC = SQRT(48) = SQRT(16 * 3) = 4 * SQRT(3)

24. In the diagram below, two concentric circles with center 0, and radii OC, OD, OGE, and ODF are drawn.

If OC = 4 and OE = 6, which relationship between the length of arc EF and the length of arc CD is always true?

Answer: (3) The length of arc EF is 1.5 times the length of arc CD.
OE is 1.5 times the length of OC because 6/4 = 1.5.
The outer circle is a dilation of the inner circle with a scale of 1.5 centered on O. That means that the lengths of the corresponding arcs will have a scale of 1.5 as well.

End of Part I.

How did you do?