August 2019 Geometry Regents Part 1 (multiple choice)

The following are some of the multiple questions from the recent June 2019 New York State Common Core Geometry Regents exam.

August 2019 Geometry, Part I

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

1. On the set of axes below, AB is dilated by a scale factor of 5/2 centered at point P.

Which statement is always true?

Answer: (2) AB | A'B'
Shape (and therefore slope) is preserved in a dilation, so the line segments are parallel.
Choice (4) is backward. If it had been 5/2(AB) = A'B', that would have been correct.

2. The coordinates of the vertices of parallelogram CDEH are C(-5,5), D(2,5), E(-1,-1), and H(-8,-1). What are the coordinates of P, the point of intersection of diagonals CE and DH?

Answer: (3) ( 3,2)
In a parallelogram, the diagonals bisect each other, so the point of intersection must be the midpoint of both lines.
The midpoint of CE is at ( (-5+-1)/2, (5+-1)/2 ), which is (-3, 2).
Checking the other line, just to be sure:
The midpoint of DH is at ( (2+-8)/2, (5+-1)/2 ), which is (-3, 2).

3. The coordinates of the endpoints of QS are Q(-9,8) and S(9,-4). Point R is on QS such that QR:RS is in the ratio of 1:2. What are the coordinates of point R?

Answer: (3) (-3,4)
A ratio of 1:2 means one-third of the way along he line.
The difference in the x-coordinates is 9 - (-9) = 18. One-third of 18 is 6, and -9 + 6 = -3.
The eliminates all but one choice.
Check the y-coordinates to be sure:
The difference in the y-coordinates is -4 - 8 = -12. One-third of -12 is -4, and 8 + -4 = 4.

4. If the altitudes of a triangle meet at one of the triangle’s vertices, then the triangle is

Answer: (1) a right triangle
There's no way for this to happen unless one of the angles is a right angle, which makes it a right triangle.

5. In the diagram below of triangle ACD, DB is a median to AC, and AB = DB

If m∠DAB = 32°, what is m∠BDC?

Answer: (3) 58°
If m∠DAB = 32°, then m∠ADB = 32° because it is an isosceles triangle.
This makes m∠DBC = 64° by the Remote Angle Theorem.
Since B is a median, AB = BC, but since AB = DB, then BC = DB, making DBC an isosceles triangle.
This makes BDC = BCD, and the pair of base angles have a sum equal to 180 - 64 = 116.
Half of 116 is 58, so m∠BDC = 58°

6. What are the coordinates of the center and the length of the radius of the circle whose equation is x² + y² = 8x - 6y + 39?

Answer: (4) center (4,-3) and radius 8
To put is into standard form (x - h)² + (y - k)² = r², you need to move the x and y terms to the left side and then Complete the Square, twice.
Remember, you need to take half of the coefficient of x (or y) and then square it to find out what to add.

x² + y² = 8x - 6y + 39
x² - 8x + y² + 6y = 39
x² - 8x + 16 + y² + 6y + 9 = 39 + 16 + 9
(x - 4)² + (y + 3)² = 64
Center (4, -3), Radius 8

Quick answer: Move the x and y term to the left, you get -8x and +6y.
This tells you that the center is going to (-1/2) of those coefficients: (4, -3)
The radius is the square root of the number on the right, and 64^2 is a really big number, so it doesn't make sense.

7. In the diagram below of parallelogram ABCD, AFGB, CF bisects <DCB, DG bisects &ADC, and CF and DG intersect at E.

If m<B = 75^o, then the measure of <EFA is

Answer: (2) 127.5°
The opposite angles of a parallelogram are congruent, and consecutive angles are supplementary. Both C and D are bisected, which cuts them in half. Since the sides are parallel, DG and CF are transversals and the alternate interior angles will be congruent.
Label the angles you know until you get to the one that you want.

Angle B is 75, so Angle DCB is 105, which is split into 52.5 and 52.5.
Alternate interior angles tells us that CFG is also 52.5. This makes Angle EFA 127.5°.

8. What is an equation of a line that is perpendicular to the line whose equation is 2y + 3x = 1?

Answer: (1) y = 2/3 x + 5/2
If you convert to slope-intercept form, you get
2y = -3x + 1
y = -3/2 x + 1/2
This is the same line, not the line perpendicular to it, which will have a slope that is the multiplicative inverse of -3/2, which is 2/3.

In this problem, the y-intercept is not important because you aren't given a specific point, and only one choice has the slope 2/3.

9. Triangles ABC and RST are graphed on the set of axes below.

Which sequence of rigid motions will prove triagnle ABC = triangle RST?

Answer: (2) a rotation of 180° centered at (1,0)
Notice the orientation of the two triangles.
A line reflection over y = x would move A to (7, -1), not to point R.
A line reflection over the x-axis would produce the mirror image of RST.
A line reflection over the x-axis followed by a line reflection over y = 1 looks okay -- until you realize that you need to reflect over x = 1 not y. Don't fall for this one.

10. If the line represented by y = - 1/4 x - 2 is dilated by a scale factor of 4 centered at the origin, which statement about the image is true?

Answer: (1) The slope is - 1/4 and the y-intercept is -8.
The slope of the line would not change. The y-intercept would move four times the distance from the origin, so -2 becomes -8.

11. Square MATH has a side length of 7 inches. Which three-dimensional object will be formed by continuously rotating square MATH around side AT?

Answer: (4) a right cylinder with a radius of 7 inches
A square would become a cylinder in three dimensions. If it is rotated about one of its sides, then the side length would become the radius.

12. Circle O with a radius of 9 is drawn below. The measure of central angle AOC is 120°.
What is the area of the shaded sector of circle O?

Answer: (4) 54π
The shaded area is 240 degrees, or 2/3 of the area of the entire circle.
A = (2/3) (pi) (9)² = 54 pi

13. In quadrilateral QRST, diagonals QS and RT intersect at M. Which statement would always prove quadrilateral QRST is a parallelogram?

Answer: (3) QR = TS and QT = RS
If the opposite sides of a quadrilateral are congruent, the figure is a parallelogram.
The others could be true for some trapezoids.

14. A standard-size golf ball has a diameter of 1.680 inches. The material used to make the golf ball weighs 0.6523 ounce per cubic inch. What is the weight, to the nearest hundredth of an ounce, of one golf ball?

Answer: (2) 1.62

Find the volume is cubic inches. Then multiply be the weight per cubic inch.
Note that you were given Diameter, and not Radius.
V = (4/3) (pi) (r)³ = (4/3)*(3.141592)(1.680/2)³ = 2.48...
Weight = (2.48)(0.6523) = 1.617 = 1.62

15. Chelsea is sitting 8 feet from the foot of a tree. From where she is sitting, the angle of elevation of her line of sight to the top of the tree is 36°. If her line of sight starts 1.5 feet above ground, how tall is the tree, to the nearest foot?

Answer: (2) 7
The distance to the tree is adjacent. The height of the tree is opposite. That means use tangent.
Tan 36 = x / 8
x = 8 * tan 36 = 5.81234...
Add 1.5 for her height while sitting: 1.5 + 5.8 = 7.3

16. In the diagram below of right triangle ABC, altitude CD intersects hypotenuse AB at D.

Which equation is always true?

Answer: (1) AD/AC = CD/BC
Short leg is to hypotenuse as short leg is to hypotenuse.
The corresponding sides are proportionals of the two smaller right triangles and the larger one.

17. A countertop for a kitchen is modeled with the dimensions shown below. An 18-inch by 21-inch rectangle will be removed for the installation of the sink.

What is the area of the top of the installed countertop, to the nearest square foot?

Answer: (4) 19
Cut it into parts: rectangle 1 + rectangle 2 - rectangle 3
2 * 8 + (5-2) * 2 - (18/12 * 21/12) = 19.375 square feet

18. In the diagram below, BC connects points B and C on the congruent sides of isosceles triangle ADE, such that triangle ABC is isosceles with vertex angle A. If AB = 10, BD = 5, and DE = 12, what is the length of BC?

Answer: (3) 8
Corresponding sides are proportional. Just remember that you need to use AD, not BD
10/15 = BC/12
15 BC = 120
BC = 8

Also, 10/15 = 2/3, and 2/3 of 12 is 8.

19. In triangle ABC below, angle C is a right angle.
[Image Omitted]
Which statement must be true?

Answer: (1) sin A = cos B
What is opposite of angle A is adjacent to angle B.
Choice (4) is only true if A and B are both 45 degrees.

20. In right triangle RST, altitude TV is drawn to hypotenuse RS. If RV = 12 and RT = 18, what is the length of SV?

Answer: (2) 15
If you sketch the figure, you will see that RT is the base of a small right triangle and RT is its hypotenuse. In the larger triangle, RST, RT is the base, and RS is the hypotenuse. So you can set up a proportion:
12 / 18 = 18 / RS
12 RS = 18 * 18
RS = 18 * 18 / 12 = 27
(Also, 12/18 reduces to 2/3, and 18 is 2/3 of 27.)

Finally, SV = RS - RV = 27 - 12 = 15.

21. What is the volume, in cubic centimeters, of a right square pyramid with base edges that are 64 cm long and a slant height of 40 cm

Answer: (3) 32,768.0
You need the height of the pyramid, not the slant height. Use the Pythagorean Theorem with the slant height as the hypotenuse and half of the side length as the base. (Use half because you only want to get to the center.)
Note: Check to see if it's a multiple of 3-4-5 to save yourself a bunch of work.
If not, or if you aren't sure, continue:

32² + h² = 40²
1024 + h² = 1600
h² = 576
h = 24

Volume = (1/3) Area of base * height
V = (1/3) * 64² * 24 = 32,768

Note 1: 24 - 32 - 40 is a multiple of 3 - 4 - 5.
Note 2: Choice (4) 54613.3... is the result if you used the slant height instead of the height. Offhand, I don't know what mistakes might lead you to getting choices (1) and (2).

22. In the diagram below, chords PQ and RS of circle O intersect at T.
[Image omitted]
Which relationship must always be true?

Answer: (4) RT X TS = PT X TQ
When two chords intersect each other, the product of their segments will be equal.
The sums (Choice (3)) usually will not be because chords of a circle can be any size, up to the length of the diameter.

23. A rhombus is graphed on the set of axes below.
[IMAGE OMITTED]
Which transformation would carry the rhombus onto itself?

Answer: (4) A reflection over the line x = 0
Hopefully, it is obvious that the y-axis divides the rhombus in half. So reflecting over the y-axis will carry it onto itself. That is, if A were (-4, 1) and C were (4, 1), then A' = C and C' = A. The other two vertices would carry onto themselves (they don't move).

The y-axis has the equation x = 0.
The line y = 0 is the x-axis, which does not divide the rhombus into two congruent figures.
Rotating around the origin would leave the image mostly below the x-axis instead of above.
Reflecting over y = 1/2x + 1 would have the rhombus stretched up and down, instead of side to side. That is, the longer diagonal would be vertical, not horizontal.

24. A 15-foot ladder leans against a wall and makes an angle of 65^o with the ground. What is the horizontal distance from the wall to the base of the ladder, to the nearest tenth of a foot?

Answer: (1) 6.3
The ladder is the hypotenuse. The ground is adjacent. This means using Cosine.
Cos 65 = x / 15
x = 15 Cos 65 = 6.339.. = 6.3

End of Part I

How did you do?

Questions, comments and corrections welcome.