Friday, July 01, 2016

Common Core Geometry, Part 1 (multiple choice), June 2016

What follows is a portion of the Common Core Geometry exam. Illustrations will be added at a later time when they become available.

Part II is posted here.

Part III and IV are posted here.

June 2016 Geometry Regents, Part I

1. A student has a rectangular postcard that he folds in half lengthwise. Next, he rotates it continuously about the folded edge. Which three-dimensional object below is generated by this rotation?

(3). The cylinder. A rectangular postcard folded over is still a rectangle. Roll it along the edge and it becomes a cylinder. (The label of a soup can is a rectangle.)

2. A three-inch line segment is dilated by a scale factor of 6 and centered at its midpoint. What is the length of its image?

(4) 18 inches. If it is centered in the middle, call the left endpoint -1.5 and the right endpoint +1.5. Multiply those by 6 and you get -9 and +9. The distance between those two points is 18.

3. Kevin's work for deriving the equation of a circle is shown below.

x2 + 4x = -(y2 - 20) STEP 1 x2 + 4x = -y2 + 20 STEP 2 x2 + 4x + 4 = -y2 + 20 - 4 STEP 3 (x + 2)2 = -y2 + 20 - 4 STEP 4 (x + 2)2 + y2 = 16
In which step did he make an error in his work?

(2) Step 4. +4 was added to the left, but -4 was added to the right.

4. Which transformation of OA would result in an image parallel to OA?

(1) a translation of two units down. Translations do not affect orientation. The lines will be parallel.

5. Using the information given below, which set of triangles can not be proven similar?

(3). One triangle has sides 2 and 8, the other has 16 and 32. They are not proportional.
Choices (1) and (4) are similar because of SAS. Choice (2) has similarity due to AA.

6. A company is creating an object from a wooden cube with an edge length of 8.5 cm. A right circular cone with a diameter of 8 cm and an altitude of 8 cm will be cut out of the cube. Which expression represents the volume of the remaining wood?

(4) (8.5)3 - (1/3) pi(4)2(8). Volume of a cube is s3. Volume of a cone is (1/3)pi(r)2(h). The diameter is 8, so the radius is 4.

7. Two right triangles must be congruent if

(3) the corresponding legs are congruent. If the legs are congruent with a right angle between them (all right angles are congruent), they are congruent due to SAS. Likewise, if the legs are congruent, then the hypotenuses are congruent, so the triangles are congruent by SSS.

8. Which sequence of transformations will map Triangle ABC onto Triangle A'B'C'?

(4) dilation and rotation. It got bigger, so there's a dilation. Its orientation has changed because of a rotation.

9. In parallelogram ABCD, diagonals AC and BD intersect at E. Which statement does not prove parallelogram ABCD is a rhombus?

(1) AC = DB. Congruent diagonals prove that a parallelogram is a rectangle, not that it is a rhombus.

10. In the diagram below of circle 0, OB and OC are radii, and chords AB, BC, and AC are drawn.

Which statement must always be true?

(2) m<BAC = (1/2)m<BOC. The size of the inscribed angle is half of the central angle. There is nothing that says that AB and AC are congruent, so BAC does not have to be isosceles, even if it looks like it in the image.

11. A 20-foot support post leans against a wall, making a 70° angle with the ground. To the nearest tenth of a foot, how far up the wall will the support post reach?

(4) 18.8. The hypotenuse is 20. The opposite is x. The angle is 70°. Opposite and hypotenuse means use sine.
sin(70) = x / 20, so x = 20 * sin(70) = 18.79... = 18.8

12. Line segment NY has endpoints N(-11,5) and Y(5, -7). What is the equation of the perpendicular bisector of NY?

(1) y + 1 = (4/3)(x + 3). The slope of the line NV is (-7 - 5) / (5 + 11) = -3/4. That means that a line perpendicular to this one has a slope of 4/3. This eliminates choices (2) and (4).
The midpoint is ( (-11+5)/2, (5-7)/2 ) = (-3, -1). Flip the signs to positive, and you get choice (1).

13. In Triangle RST shown below, altitude SU is drawn to RT at U.

If SU = h, UT = 12, and RT = 42, which value of h will make Triangle RST a right triangle with LRST as a right angle?

(2) 6*SQRT(10). (6 times radical 10). Notice that they give RT and not RU. RU = 42 - 12 = 30.
h2 = (30)(12) = 360
h = Sqrt(360) = Sqrt(36)*Sqrt(10) = 6*Sqrt(10).

14. In the diagram below, Triangle ABC has vertices A(4,5), B(2,l), and C(7,3).

What is the slope of the altitude drawn from A to BC?

(4) -5/2. The altitude from A to BC has a slope that is the inverse reciprocal of the slope of BC, because the altitude is perpendicular to BC. The slope of BC is 2/5 (up 2 boxes and right 5). The inverse reciprocal is -5/2.
Note that (-5/2)(2/5) = -1, which is true for inverse reciprocals.

15. In the diagram below, Triangle ERM ~ Triangle JTM.

Which statement is always true?

(4) tan E = TM/JM. Angle E corresponds to angle J. Tan J = opposite / adjacent = TM / JM.

16. On the set of axes below, rectangle ABCD can be proven congruent to rectangle KLMN using which transformation?

(3) reflection over the x-axis.

17. In the diagram below, DB and AF intersect at point C, and AD and FBE are drawn. If AC= 6, DC= 4, FC = 15, m<D = 65°, and m<CBE = ll5°, what is the length of CB?

(1) 10. If CDE = 115, then CDF = 65°, so the triangles are similar. (The vertical angles are congruent also.) This makes the corresponding sides proportional.
4 / 6 = x / 15
6x = (4)(15) = 60
x = 10

18. Seawater contains approximately 1.2 ounces of salt per liter on average. How many gallons of seawater, to the nearest tenth of a gallon, would contain 1 pound of salt?

(2) 3.5. Conversion: 1 pound = 16 ounces. 1 liter = 0.264 gallon. (This comes from the reference table in the back of the book.) Compare ounces to gallons with ounces to gallons.
1.2 / .264 = 16 / x
1.2x = (.264)(16) = 4.224
x = 3.52 = 3.5

19. Line segment EA is the perpendicular bisector of ZT, and ZE and TE are drawn.

Which conclusion can not be proven?

(2) Triangle EZT is equilateral. EZT is isosceles, but not necessarily equilateral.

20. A hemispherical water tank has an inside diameter of 10 feet. If water has a density of 62.4 pounds per cubic foot, what is the weight of the water in a full tank, to the nearest pound?

(1) 16,336/ Density equals mass divided by Volume, in this case pounds per cubic foot. So mass(weight) = Density * Volume.
Volume = (1/2) * (4/3) * pi * r3 = (4 / 6) * pi * 53 = 261.799...
Mass = 62.4 * 261.799 = 16336.2576 = 16,336 pounds.

21. In the diagram of Triangle ABC, points D and E are on AB and CB, respectively, such that AC || DE.

If AD = 24, DB = 12, and DE = 4, what is the length of AC ?

(2) 12. The triangles are similar, so DB/DE = AB/AC. AB = AD + DB = 24 + 12 = 36.
36 / x = 12 / 4
12x = 144
x = 12

22. Triangle RST is graphed on the set of axes below.

(3) 45. The area of a triangles is 1/2*b*h. The base and height are perpendicular to each other. It is easy to see that angle S is a right angle by comparing the slopes. SR has a slope of 2 (you can see this by counting boxes). ST has a slope of -1/2. Multiply 2*(-1/2) = -1, so the lines are perpendicular, and RST is a right triangle.

The length of RS = SQRT(32 + 62) and the length of ST is SQRT(122 + 62).
Multiply 1/2 * SQRT(9 + 36) * Sqrt(144 + 36) = 45. (If you got 90, you forgot the 1/2.)

23. The graph below shows AB, which is a chord of circle 0. The coordinates of the endpoints of AB are A(3,3) and B(3,-7). The distance from the midpoint of AB to the center of circle 0 is 2 units.

What could be a correct equation for circle O?

(1) (x - 1)2 + (y + 2)2 = 29. The center of the circle must be at (1, -2) or (5, -2). The signs must be flipped in the equation of the circle, so any choice with +1, +5 or -2 can be eliminated. Choices (2) and (3) are out. The diameter has to be longer than the chord, so the radius has to be longer than 5.

24. What is the area of a sector of a circle with a radius of 8 inches and formed by a central angle that measures 60°?

(3) 32 * pi / 3. A = (60 / 360) * pi * (8)22 = 64 * pi / 6 = 32 * pi / 3

End of Part I