Here are the questions, with answers and explanations, for the New York State Geometry Regents (not Common Core) exam, Part 2. There were 6 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
29. The image of after a reflection through the origin is R'S'. If the coordinates of the endpoints of are R(2,-3) and S(5,1), state and label the coordinates of R' and S '.[The use of the set of axes below is optional.]
In a reflection about the origin, P(x, y) -> P'(-x, -y)
R(2, -3) -> R'(-2, 3)
S(5, 1) - > S'(-5, -1)
30. A paper container in the shape of a right circular cone has a radius of 3 inches and a height of 8 inches. Determine and state the number of cubic inches in the volume of the cone, in terms of π.
The Volume of a Right Circular Cone is found using, V = (1/3)Bh, where B is the area of the base. The area of the Base is πr2, so V = (1/3)πr2h = (1/3)π(3)2(8) = 24π.
31. In isosceles triangle RST shown below, RS = RT, M and N are midpoints of RS and RT, respectively, and MN is drawn. If MN = 3.5 and the perimeter of triangle RST is 25, determine and state the length of NT.
MN is the midsegment, so ST is twice as long. Since MN = 3.5, ST = 7. RST is isosceles and the perimeter is 25, so x + x + 7 = 25. 2x + 7 = 25, 2x = 18, x = 9.
RS and RT have lengths of 9. N is the midpoint of RT, so NT is 1/2 of RT. NT = 4.5
32. In the diagram below, ABC is equilateral. Using a compass and straightedge, construct a new equilateral triangle congruent to ABC in the space below. [Leave all construction marks.]
Make a point on the lower portion of the page. Call it P. Use the compass to measure the length of AB. Make a little arc. Go back to the point you made and, without changing the compass, make an arc. Make a point on the arc. Call it Q. Use the straightedge to make PQ. Still not changing the compass, make an arc from point P above the middle of the line segment. Make a similar arc from point Q. Where the two arcs intersect, make a point. Call it R. Use the straightedge to make PR and QR. PQR is an equilateral triangle.
For a visual reference, look at Method 1 on this MathBits page.
33. Write an equation of the line that is perpendicular to the line whose equation is 2y = 3x + 12 and that passes through the origin.
In slope-intercept form, the given line is y = (3/2)x + 6. A line perpendicular to it would have to have slope = -2/3. If it goes through the origin, the y-intercept is 0.
Therefore, y = (-2/3)x.
34. Rectangle KLMN has vertices K(0,4), L(4,2), M(1,-4), and N(-3,-2). Determine and state the coordinates of the point of intersection of the diagonals.
The diagonals of a rectangle bisect each other, so you just need to find the midpoint of either of the diagonals.
( (0 + 1)/2, (4 + (-4))/2 ) = (1/2, 0) or
( (4 + (-3))/2, (2 + -2)/2 ) = (1/2, 0).
That's it for Part II. Parts III and IV are coming soon. I hope.