Below are the questions with answers and explanations for Part 1 of the New York State Geometry Regents (not Common Core) exam for January 2016.
1. What is the equation of a circle with its center at (5,-2) and a radius of 3?
(2) (x - 5)2 + (y + 2)2 = 9. Flip the signs on (h, k) and square the radius.
2. In the diagram below, <ABC is inscribed in circle 0.
The ratio of the measure of <ABC to the measure of arc AC is
(2) 1:2. The ratio of the inscribed angle to the arc it intercepts is 1:2. In other words, the arc is twice the size of the inscribed angle.
3. In the diagram below of rectangle RSTU, diagonals RT and SU
intersect at 0.
If RT = 6x + 4 and SO = 7x - 6, what is the length of US?
(3) 16. The diagonals of a rectangle are congruent so RT = US. The diagonals of a rectangle also bisect each other, so the length of UO = SO, and US = 2(SO). So US = 14x - 12.
8x = 16
x = 2
US = 14(2) - 12 = 28 - 12 = 16
4. How many points are 3 units from the origin and also equidistant from both the x-axis and y-axis?
(4) 4. Three units away from the origin is a circle. Equidistant from the x-axis and y-axis are two diagonal lines (y = x and y = -x). The circle intersects each line twice. Four points.
5. The converse of the statement "If a triangle has one right angle, the triangle has two acute angles" is
(1) If a triangle has two acute angles, the triangle has one right angle. Note that the converse does not have the same Truth value as the statement.
6. The surface area of a sphere is 2304(pi) square inches. The length of a radius of the sphere, in inches, is
(2) 24. The formula for the Surface Area of a sphere is 4(pi)r2 = 2304(pi). Dividing by 4 gives you 576. Take the square root, and the radius is 24.
7. As shown in the diagram below of triangle ABC, BC is extended through D,
m<A = 70, and m<ACD = 115.
Which statement is true?
(4) AC < AB. Angle ACB is 65o and angle B is 45o. So AC < AB < BC
8. In trapezoid LMNO below, median PQ is drawn.
If LM = x + 7, ON = 3x + 11, and PQ = 25, what is the value of x?
(3) 8. The length of the median of a trapezoid is half the sum of the base heights. That is, it's is the average of the length of the two bases. So x + 7 + 3x + 11 = 25 * 2.
4x + 18 = 50
4x = 32
x = 8. The three lines would be 15, 25 and 35.
9. Points A and B are on line L. How many points are 3 units from line L and also equidistant from A and B?
(2) 2. Second Locus question. Equidistant from line L are two lines parallel to L. Equidistant to points A and B is the perpendicular bisector of AB, which will intersect each of the two parallel lines.
10. The lines whose equations are 2x + 3y = 4 and y = mx + 6 will be perpendicular when m is
(3) 3/2. The slope of the perpendicular line is the inverse reciprocal of the first line. Subtract 2x from both sides and divide by three, and the slope of the first line is -2/3. The inverse reciprocal is 3/2.
11. As shown in the diagram below, M, R, and T are midpoints of
the sides of triangle ABC.
If AB = 18, AC 14, and BC 10, what is the perimeter of quadrilateral ACRM?
(1) 35. MR is a midsegment, so it is parallel to AC and half of its size. AM is 9, AC is 14, CR is 5 and RM is 7. Add 9 + 14 + 5 + 7 = 35.
12. In the diagram below, ABC || DEFG. Transversal BHE and line segment HF are drawn.
If m<HFG = 130 and m<EHF = 70, what is m<ABE?
(3) 60. Angle HFG is an exterior angle to triangle EFH. Angles EHF and HEF are remote angles with a sum of 130 degrees. Angle EHF = 70, so HEF = 60. Angle ABE and HEF are alternate interior angles, so they are congruent.
13. The graphs of the lines represented by the equations y = (1/3)x + 7 and y = -(1/3)x - 2 are
(4) intersecting, but not perpendicular. Second question about the slopes of perpendicular lines. The product of (1/3) and (-1/3) is NOT -1, so they are not perpendicular.
14. Which graph represents a circle whose equation is (x + 3)2 + (y - 1)2 = 4?
(1). The circle with the center at (-3, 1) with a radius of 2. Second question about the equation of a circle.
15. In triangle ABC, m<CAB = 2x and m<ACB = x + 30. If AB is extended through point B to point D, m<CBD = 5x - 50. What is the value of x?
(3) 40. Make a diagram if it helps.
Angles CAB and ACB are remote angles to exterior angle CBD. So the sum of 2x + x + 30 = 5x - 50.
3x + 30 = 5x - 50
80 = 2x
40 = x.
The angles of ABC are 80, 70 and 30. The exterior angle is 150.
16. In circle O shown below, chord AB and diameter CD are parallel,
and chords AD and BC intersect at point E.
Which statement is false
(2) BE = CE. The two triangles, ABE and CDE, will be similar but not congruent, so point E is NOT the midpoint of BC. If you draw in AC and BD, you will have an isosceles trapezoid. The diagonals of an isosceles trapezoid do NOT bisect each other.
17. When the transformation T(2,-1) is performed on point A, its image is point A'(-3,4). What are the coordinates of A?
(2) (-5, 5). (-5 + 2 = -3, 5 - 1 = 4).
18. If the sum of the interior angles of a polygon is 1440°, then the polygon must be
(2) a decagon. I should go as far as to say that you should recognize the number if you did any of these problems, but I'll explain anyway. The formula is (n - 2) * 180 = 1440, so n - 2 = 8, n = 10. Ten sides is a decagon. If you didn't remember the formula, you could have started with a triangle and kept adding 180 degrees until you hit 1440.
19. In triangle ABC shown below, medians AD, BE, and CF intersect at
If CR = 24 and R F= 2x - 6, what is the value of x?
(1) 9. Medians meet at the centroid, and the distance from the angle to the centroid is twice the distance from the centroid to the midpoint of the opposite side.
CR is twice the size of RF, so 2x - 6 = 12. Then 2x = 18 and x = 9.
20. Which equation represents a line that passes through the point (-2,6) and is parallel to the line whose equation is 3x - 4y = 6?
(3) -3x + 4y = 30. Parallel lines have the same slope. The slope of each line is 3/4.
-3(-2) + 4(6) = 6 + 24 = 30. Check. (Note that choice (3) was the only possibility based on slope alone, but it's a good idea to check the point anyway. Just in case.)
21. The bases of a right prism are triangles in which triangle MNP = triangle RST. If MP = 9, MR = 18, and MN = 12, what is the length of NS?
(4) 18. I had to read that one twice to see if this was a trick question. But that depends on what you think is a trick question. If it is a prism, then the bases are parallel to each other. That makes MR, NS and PT are the same size, which is 12. If you set up ratios or tried the Pythagorean Theorem, you were incorrect.
22. Triangle ABC has the coordinates A(3,0), B(3,8), and C(6,6). If triangle ABC is reflected over the line y = x, which statement is true about the image of triangle ABC?
(1) One point remains fixed. If you reflect it over y = x, the point (6, 6) will remain at (6, 6) because it is on the line y = x. (e.g., 6 = 6) The size remains the same. The orientation changes. And the final statement, besides not being true, has nothing to do with the image or the reflection.
23. A right circular cone has a diameter of 10(sqrt(2)) and a height of 12. What is the volume of the cone in terms of pi?
(1) 200 pi. The volume is (1/3)(pi)r2h = (1/3)(pi)(5(sqrt(2))(12) = 200 pi.
Choice (2) is if you forgot the (1/3). Choices (3) and (4) if you forgot to halve the diameter before squaring.
24. Which statement is not always true when triangle ABC = triangle XYZ?
(2) CA = XY. If you didn't get this one correct, it might be because your teacher wasn't careful enough in examples. Or because you didn't pay attention the time they mentioned that when written properly, each letter, in order, should correspond. So A to X, B to Y, C to Z.
25. If two sides of a triangle have lengths of (1/4) and (1/5), which fraction can not be the length of the third side?
(4) 1/2. The third side cannot be bigger than 1/4 + 1/5 nor smaller than 1/4 - 1/5. One half is greater than 9/20. Also, 1/2 = 1/4 + 1/4 which is bigger than 1/4 + 1/5.
26. In the diagram below of triangle ABC, CDA, CEB, DE || AB, DE = 4,
AB = 10, CD = x, and DA = x + 3.
What is the value of x?
(4) 6. Set-up the proportion CD/DE = CA/AB
x / 4 = (2x + 3) / 10. (Don't forget to add x and x + 3.)
10x = 8x +12
2x = 12
x = 6.
27. Given: AE bisects BD at C
AB and DE are drawn
<ABC = <EDC
Which statement is needed to prove ABC = EDC using ASA?
(3) <BCA = <DCE. You are given a pair of angles. The bisecting gives the included side. The two vertical angles are the other pair of angles.
28. In the construction shown below, CD is drawn.
In triangle ABC, CD is the
(2) median to side AB. From the marks, a perpendicular bisector of AB was constructed. Point D is therefore the midpoint of AB, making CD the median.
End of Part I.