August 2015 Geometry (Common Core) Regents: Part 2, Open-Ended

Here are the questions, with answers and explanations, for the New York State Geometry (Common Core) Regents exam, Part 2. There were 7 questions, each worth 2 credits. Partial credit may be earned for correct work on a problem without a solution, or for a problem with a solution that contains one computational or conceptual error. All work must be shown. In general, a correct answer without any work is worth 1 credit, unless that answer is given as a choice and an explanation is required.

Link to Part 1

Part 2

25. A wooden cube has an edge length of 6 centimeters and a mass of 137.8 grams. Determine the density of the cube, to the nearest thousandth.
State which type of wood the cube is made of, using the density table below.

Density = mass / Volume. Volume of a cube = (edge length)³.
Therefore, Density = mass / (edge)³ = 137.8 / (6)³ = 0.63796...

This is the approximate Density of Ash wood, according to the table.

26. Construct an equilateral triangle inscribed in circle T shown below. [Leave all construction marks.]

Using the straightedge, make a diameter in circle T. Label one of the points. (I used S.) Use the compass to measure radius ST. Make a circle around point S. Label the two points of intersection (A and B). Use the compass to measure the distance from A to B, make a small arc. Use the straightedge to draw chord AB. Label the other end of the diameter (C). Without changing the compass, put one end at A and make a small arc that crosses C. Without changing the compass, put one end at B and make a small arc that crosses C. Use the straightedge to draw AC and BC.

(Image to come.)

27. To find the distance across a pond from point B to point C, a surveyor drew the diagram below. The measurements he made are indicated on his diagram.

Use the surveyor’s information to determine and state the distance from point B to point C, to the nearest yard.

Make a proportion: 230/120 = (230 + 85) / x
230 x = (120)(315)
230 x = 37800
x = 164.3478...
x = 164 yards

28. In parallelogram ABCD shown below, diagonals AC and BD intersect at E.

Prove: ∠ACD = ∠CAB

Statement: Reason
ABCD is a parallelogram: given
DC || AB, DA || CB: Opposite sides of a parallelogram are parallel.
Angle ACD = Angle BAC: If two parallel lines are cut by a transversal, the alternate interior angles are congruent.

29. Triangles RST and XYZ are drawn below. If RS = 6, ST = 14, XY = 9, YZ = 21, and ∠S = ∠Y, is triangle RST similar to triangle XYZ? Justify your answer.

If the triangles are similar than 6/14 = 9/21. Cross-multiply you get 126 = 126. (Alternatively, you could reduce both sides to 2/7.) Because a pair of corresponding sides are proportional and the included angles between them are congruent, then by SAS, RST ~ XYZ.

30. In the diagram below, triangle ABC and triangle XYZ are graphed.

Use the properties of rigid motions to explain why triangle ABC = triangle XYZ.

In a rotation of 180 degrees, the image of point (x, y) is (-y, -x). This transformation is a rotation and when rotating size and shape are preserved. So ABC = XYZ

31. The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE:EF = 2:3.

16 - 1 = 15. 15 * (2/5) = 6. 1 + 6 = 7
14 - 4 = 10. 10 * (2/5) = 4. 4 + 4 = 8. E(7,8).

End of Part II