January 2018 Common Core Geometry Regents, Part 2 (open-ended)

The following are some of the multiple questions from the recent January 2018 New York State Geometry Regents exam.
The questions and answers to Part I can be found here.

January 2018 Geometry, Part II

Each correct answer is worth up to 2 credits. Partial credit is available. Work must be shown. Correct answers without work receive only 1 point.

25. Given: Parallelogram ABCD with diagonal AC drawn

Prove: triangle ABC = triangle CDA

Answer: You can give either a paragraph or two-column proof. However, when writing a paragraph, you still need to remember to have all the statements and reasons.
ABCD is a parallelogram; Given.
AB = CD, AD = BC; Opposite sides of a parallelogram are congruent.
AC = AC; Reflexive property
triangle ABC = triangle CDA; SSS

26. The diagram below shows circle 0 with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle 0. [Leave all construction marks.]

Answer: Strategy: if you draw a perpendicular bisector between A and B, you will get a vertical line through the center of the circle O. Where the line intersects the circle will be the other two vertices of the square. Use your straightedge to draw the square.

27. Given: Right triangle ABC with right angle at C
If sin A increases, does cos B increase or decrease? Explain why

Answer: Cos B will increase because the sin A = cos B. This is only a partially correct response. You need to back it up with a definition or explanation.
Sine is the ratio of opposite over hypotenuse (you can write this as a fraction.)
Cosine is the ratio of adjacent over hypotenuse.
The side that is opposite angle A is the same side that is adjacent to angle B.
Therefore sin A and cos B are the same ratio.

28. In the diagram below, the circle has a radius of 25 inches. The area of the unshaded sector is 500 pi in².

Determine and state the degree measure of angle Q, the central angle of the shaded sector.

Answer: Find the area of the circle. Subtract 500pi from it. Compare the result to the original as a fraction. Multiply that fraction by 360 degrees.
A = (pi)(r)²
A = (pi)(25)²
A = 625pi
The shaded area is 625pi - 500pi = 125pi
The fraction of the circle that's shaded is (125pi)/625pi) = 125/625 = 1/5
1/5 (360) = 72 degrees.

29. A machinist creates a solid steel part for a wind turbine engine. The part has a volume of 1015 cubic centimeters. Steel can be purchased for $0.29 per kilogram, and has a density of 7.95 g/cm3.
If the machinist makes 500 of these parts, what is the cost of the steel, to the nearest dollar?

Answer: Find the mass of one part, and multiply it by 500 to get the mass of 500 parts in grams. Divide by 1000 to change it into kilograms. Then multiply the mass by $0.29 per kilogram.
D = m/V
7.95 = m/1015
m = 1015(7.95) = 8069.25
8069.25 * 500 = 4,034,625
4,034,625 / 1000 = 4,034.625
4,034.625 * 0.29 = 1,170.04125
$1,170 (to the nearest dollar)
Note: If you do all this work and forget to round to the nearest dollar, you will lose 1 of the 2 available points.
That stinks, but there's nothing you can do about that.

30. In the graph below, triangle ABC has coordinates A(-9,2), B(-6,-6), and C(-3, -2), and triangle RST has coordinates R(-2,9), S(5,6), and T(2,3).

Is triangle ABC congruent to triangle RST? Use the properties of rigid motions to explain your reasoning.

Answer: They are not congruent. If they were congruent then you could map ABC onto RST with a series of rigid motions. If you reflect ABC over the line y = -x, A would map to R and C would map to T, but B would map to (6, 6), not to S(5, 6). So the triangles are not congruent.

You could also use the distance formula to show that the lengths of the sides of the triangles are not the same, but you still need to mention the properties of rigid motions to get full credit.

31. Bob places an 18-foot ladder 6 feet from the base of his house and leans it up against the side of his house. Find, to the nearest degree, the measure of the angle the bottom of the ladder makes with the ground.

Answer: Sketch a little picture, if it helps. The ladder is the hypotenuse, and the distance to the house is the adjacent side.
Use cos x = adj / hyp
cos x = 6 / 18
x = cos^-1(6/18) = 70.5287793655... = 71 degrees.


End of Part II

How did you do?
Questions, comments and corrections welcome.