Wednesday, August 19, 2015

August 2015 Geometry (Common Core) Regents: Part 1, Multiple-Choice

Here are the questions, with answers and explanations, for the New York State Geometry (Common Core) Regents exam. There were 24 questions, each worth 2 credits. No partial credit. No work needed to be shown (but it would still be a good idea to work out the answers, even if no one will see it). According to the conversion chart for this exam, 33 credits are needed to achieve a scaled score of 65. While that number is a little higher than the Common Core Algebra 1 exam, it is significantly lower than the raw score required on recent non-Common Core Geometry Regents. So seventeen correct out of the twenty-four questions asked would be enough to pass the exam.

Part 1

1. A parallelogram must be a rectangle when its

(2) diagonals are congruent. Choices (1) and (4) is not true for all rectangles. Choice (3) applies to all parallelograms.

2. If triangle A'B'C' is the image of triangle ABC, under which transformation will the triangles not be congruent?

(3) dilation centered at the origin with scale factor 2. The center doesn't matter. When you dilate, the size changes. Therefore, the image cannot be congruent to the original.

3. If the rectangle below is continuously rotated about side w, which solid figure is formed? (image omitted)

(4) cylinder. Although the rectangle is flat on the paper, it is being rotated into 3-D space, and the figure formed will be a three-dimensional solid. Imagine the rectangle is a little flag and side w is attached to the flag pole. If you spin the pole, the flag goes around in a circle. The circle on top will be the same as the circle on the bottom, directly below it. The space between them is a cylinder.

4. Which expression is always equivalent to sin x when 90 < x < 90o?

(1) cos(90o - x). Sine and cosine are complementary. In a right triangle the sine of one acute angle is the same as cosine of the other acute angle. Consider the definitions opposite/hypotenuse and adjacent/hypotenuse. What is opposite on angle is adjacent to the other.

5. In the diagram below, a square is graphed in the coordinate plane. (image omitted).
A reflection over which line does not carry the square onto itself?

(1) x = 5. In choice (1), the side of the square on the vertical line x = -1 would end up at x = 11. Choices (3) and (4) are the diagonals of the square; flipping over these will cause the square to remain in place. Choice (2) is the horizontal line that divides the square in half; reflecting over this line will put the square back into its original place.

6. The image of triangle ABC after a dilation of scale factor k centered at point A is triangle ADE, as shown in the diagram below. (image omitted)
Which statement is always true?

(4) BC || DE. In a dilation, size changes, but orientation does not. Therefore, the corresponding parts of the two objects will have the same slopes, making them either parallel or part of the same line. Choices (1) and (3) are true only when the scale factor is 2. Choice (2) could only be true if we knew that AB is perpendicular to BC, but that is not stated.

7. A sequence of transformations maps rectangle ABCD onto rectangle A"B"C"D", as shown in the diagram below. (image omitted)
Which sequence of transformations maps ABCD onto A'B'C'D' and then maps A'B'C'D' onto A"B"C"D"?

(1) a reflection followed by a rotation. First step, reflect over the x-axis. Second step, rotate 90 degrees (counterclockwise) about the origin. If it had bee a translation, A' would have been above D' (for example).

8. In the diagram of parallelogram FRED shown below, ED is extended to A, and AF is drawn such that AF = DF. (image omitted)
If m<R = 124°, what is m<AFD?

(3) 68°. AF = DF means that it is an isosceles triangle and the base angles are the same. Angle FDE is congruent to angle R, so it is also 124°. Subtract 180 - 124 = 56. Solve x + 56 + 56 = 180, x = 68.

9. If x2 + 4x + y2 - 6y - 12 = 0 is the equation of a circle, the length of the radius is

Complete the squares:

x2 + 4x + y2 - 6y - 12 = 0
x2 + 4x + 4 + y2 - 6y + 9 - 12 = 0 + 4 + 9
(x + 2)2 + (y - 3)2 - 12 + 12 = 13 + 12
(x + 2)2 + (y - 3)2 = 25
Thus r2 = 25, therefore r = 5.

10. Given MN shown below, with M(-6,1) and N(3,-5), what is an equation of the line that passes through point P(6,1) and is parallel to MN? (image omitted)

(1) y = (-2/3)x + 5. Parallel means that it has the same slope, which in this case is negative. Only two choices are negative and they have the same slope, so that is it. You can substitute (6, 1) in both equations and you will see that choice (1) is a solution and choice (2) is not.

OR Calculate the slope of the line, either with the formula or counting the boxes, (-5 - 1)/(3 - -6) = -6/9 = -2/3. Eliminate choices (3) and (4). Use (6, 1) to solve this equation

(1) = (-2/3)(6) + b
1 = -4+ b
b = 5

11. Linda is designing a circular piece of stained glass with a diameter of 7 inches. She is going to sketch a square inside the circular region. To the nearest tenth of an inch, the largest possible length of a side of the square is

(2) 4.9. The largest square inside a circle will be one that is inscribed in the circle (vertices on the circle), which will have a diameter for its diagonal. That diagonal forms two right triangles with legs that are equal in length and the hypotenuse equal to the diameter, which is 7.

So s2 + s2 = 72
2s2 = 49
s2 = 24.5
s = 4.9497...

12. In the diagram shown below, AC is tangent to circle 0 at A and to circle P at C, OP intersects AC at B, OA = 4, AB= 5, and PC= 10. (image omitted)
What is the length of BC?

(3) 12.5. Triangles OAB and PCB are similar by AA because they have two pairs of congruent angles: the right angles and the vertical angles. Therefore, the corresponding sides are proportional: 4/10 = 5/x. x = 50/4 = 12.5.

13. In the diagram below, which single transformation was used to map triangle A onto triangle B? (image omitted)

(2) rotation. B is the image of A after a 90o counterclockwise rotation.

14. In the diagram below, triangle DEF is the image of triangle ABC after a clockwise rotation of 180° and a dilation where AB = 3, BC = 5.5, AC= 4.5, DE= 6, FD= 9, and EF = 11. (image omitted)
Which relationship must always be true?

(4) m<B/m<E = m<C/m<F. The scale factor is 2, but that has to do with the sides of the triangle, not the angles. The angles are congruent. Choice (3) does not relate corresponding angles in the proportion.

15. In the diagram below, quadrilateral ABCD is inscribed in circle P. (image omitted)
What is m<ADC?

(3) 108o. The opposite angles of an inscribed quadrilateral are supplementary because they will intercept two arcs (one major, one minor; or two semicircles) that form a complete circle of 360 degrees. Since angle ABC is 72 degrees, then ADC is 108. Likewise, angle BCD is 70.

16. A hemispherical tank is filled with water and has a diameter of 10 feet. If water weighs 62.4 pounds per cubic foot, what is the total weight of the water in a full tank, to the nearest pound?

(1) 16,336. The radius is half the diameter; half of 10 is 5. Use the formula for Volume of a sphere and halve it to find a hemisphere: V = (1/2)(4/3)(pi)(r)3 = 261.799...
Multiply 261.799 * 62.4 = 16336.28.., which rounds down. If you're off by a little bit (but didn't get one of the other answers, it's because you didn't use enough decimal places.
Note: Choice (2) is if you forget the 1/2 for the hemisphere. Choice (3) is if you used 10, not 5. Choice (4) is if you made both mistakes. Given this, you could have reasoned out the correct answer through approximation.

17. In the diagram below, triangle ABC ~ triangle ADE. (image omitted)
Which measurements are justified by this similarity?

(4) AD = 2, AB = 6, AE = 5, and AC =15. AD/AB = AE/AC. The corresponding sides are proportional. The scale factor is 3.

18. Triangle FGH is inscribed in circle O, the length of radius OH is 6, and FH = OG. (image omitted)
What is the area of the sector formed by angle FOH?

(3) 6*pi. FH = OG. OG = OH. OH = 6. Therefore, FH = 6. Also, OF = 6, because it is another radius. Triangle FOH is equilateral, which means that all of its angles are 60 degrees. The central angle of 60 degrees marks off 60/360 = 1/6 of the circle. The area of 1/6 of the circle is (1/6)pi*r2 = (1/6)(6)2*pi = 6 pi.

19. As shown in the diagram below, AB and CD intersect at E, and AC || BD. (image omitted)
Given triangle AEC ~ triangle BED, which equation is true?

(2) (AE/BE) = (AC/BD). Given that the triangles are similar, their corresponding sides are proportional. Choice (2) has the corresponding sides in the correct order.

20. A triangle is dilated by a scale factor of 3 with the center of dilation at the origin. Which statement is true?

(1) The area of the image is nine times the area of the original triangle. The area of a triangle is one half times base times height: (1/2) b h. The area of the image is (1/2) (3b) (3h) = (9)(1/2) b h.

21. The Great Pyramid of Giza was constructed as a regular pyramid with a square base. It was built with an approximate volume of 2,592,276 cubic meters and a height of 146.5 meters. What was the length of one side of its base, to the nearest meter?

(4) 230. Volume is (1/3)(Area of the base)(height).

2,592,276 = (1/3)(Area)(146.5)
Area = 53084.150...
Side = square root (53084.150)
Side = 230.400...

22. A quadrilateral has vertices with coordinates (-3,1), (0,3), (5,2), and (-1,-2). Which type of quadrilateral is this?

(4) trapezoid. Find the slopes of the lines. One pair of equal slopes means trapezoid. Two pairs means some type of parallelogram (the other three choices). First line is (3-1)/(0-(-3)) = 2/3. Second line is (2-3)/(5-0) = -1/5. Third line is (-2-2)/(-1-5)= -4/-6 = 2/3. Fourth line is (1-(-2))/(-3-(-1)) = 3/-2 = -3/2. One pair of parallel sides is a trapezoid.

23. In the diagram below, triangle ABE is the image of triangle ACD after a dilation centered at the origin. The coordinates of the vertices are A(O,O), B(3,0), C(4.5,0), D(0,6), and E(0,4). (image omitted)
The ratio of the lengths of BE to CD is

(1) 2/3. You don't need to find the lengths of BE or CD to solve this. You only need the scale factor. AE/AD = 4/6 = 2/3. That's it.

24. Line y = 3x - 1 is transformed by a dilation with a scale factor of 2 and centered at (3,8). The line's image is

(4) y = 3x - 1. Because (3, 8) is a solution to the equation, it is a point on the line. If a line is dilated but goes through the center of the dilation, the line is unchanged. A line segment would have its length changed, but a line is infinite in length, so it is unaffected.

That's the end of Part 1. Part 2 will hopefully be uploaded soon, and a link will be added here.

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Anonymous said...

Is there another way to figure the problem about Linda and her circular glass?

(x, why?) said...

Do you have something in mind?

It's pretty straightforward. There's a square in the circle. The diagonal will be the diameter of the circle. It's a 45-45-90 right triangle.

You could just divide 7 by sqrt(2) if you recognize the properties of a 45-45-90 triangle. You'll get the same answer.