Wednesday, August 13, 2014

August 2014 Geometry Regents, Part 1 (Multiple Choice)

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I just spent a few hours grading the Geometry Regents exams at my old school. It was described as "doable". I think that the students who came to my tutoring classes the past few days will be happy ... assuming that they were all working out their own problems and not just copying my answers.

A few observations: the proof was pretty straightforward, with a choice of AAS or ASA, as long as you backed up the one you used; the locus question was very unusual in that you had to give an equation of the circle locus and the line was neither vertical nor horizontal; the construction question had you bisecting multiple times (and using your ruler and drawing the curves later didn't count). These issues with the Open-Ended problems with be discussed tomorrow (I hope).

Here are the multiple-choice questions from Part I. Keep in mind, that I have to type all of these, so the rest of the test may not show up on my blog as quickly as you may like. Questions are always welcome. Likewise, because I've been asked to hurry with this, there are no diagrams included. They may get added at a later time.

1. A rectangular prism is shown in the diagram below. [DIAGRAM IS BOX-SHAPED WITH DIAGONALS ON TOP AND BOTTOM DRAWN.]
Which pair of line segments would always be both congruent and parallel?

(4) DB and HF are parallel. Of the other three choices, two pair intersect and one pair is skew.

2. In parallelogram QRST, diagonal QS is drawn. Which statement must always be true?

(3) Triangle STQ is congruent to QRS. The diagonal of a parallel creates two congruent triangles by SSS. (Proof left as an exercise to the reader.)

3. In the diagram below of circle O, diameter AB and chord CD intersect at E. [DIAGRAM IS A CIRCLE AS DESCRIBED.]
If AB is perpendicular to CD, which statement is always true?

(4) Arc CB is congruent to arc BD. (Also arcs AC and AD are congruent, in case you were wondering.)

4. What is an equation of the line that passes through (-9, 12) and is perpendicular to the line whose equation is y = 1/3 x + 6?

(2) y = -3x - 15. If it is Perpendicular then the slopes are negative reciprocals. The slope of the new line must be -3. (Eliminate two choices.) Plugging in (-9, 12) into choice (2) shows that it is a point on that line. Likewise, substitution into equation (4) shows that it is not correct.

5. In the diagram below, under what transformation is triangles X'Y'Z' the image of triangle XYZ? [DIAGRAM SHOWS A CO-ORDINATE PLANE WITH A TRIANGLE IN QUADRANT I AND A CONGRUENT, BUT *ROTATED* TRIANGLE IN QUADRANT IV.]

(3) It's a rotation of 90 degrees clockwise.

6. What is the solution of the system of equations y - x = 5 and y = x2 + 5?

(1) You can check very quickly that (0, 5) is a solution. From the choices, the other solution has to be (1, 6) or (-1, 6). Check (1, 6); it works.

7. In the diagram below, parallelogram ABCD has vertices A(1, 3), B(5, 7), C(10, 7) and D(6, 3). Diagonals AC and BD intersect at e.

What are the coordinates of point E?

(3) (5.5, 5) The diagonals of a parallelogram bisect each other, so the coordinates of point E is the midpoint of either diagonal.

After a reflection over the y-axis, the image of triangle ABC is triangle A'B'C'. Which statement is not true?

(4) AC will not be parallel to A'C' after the reflection. A'C' will have a positive slope.

9. What is an equation of circle O shown in the graph below? [DIAGRAM SHOWS A CO-ORDINATE PLANE WITH A CIRCLE CENTERED AT (-2,4) WITH RADIUS 4.]

(4) Flip the signs, square the radius.

10. In the diagram below of right triangle ABC, an altitude is drawn to the hypotenuse AB.

Which proportion would always represent a correct relationship of the segments?

(3) The Right-Triangle Altitude Theorem. Altitude z is used twice in the proportion.

Which quadrilateral best classifies ABCD?

(3) It's a rhombus, but not a square.

12. Circle O is represented by the equation (x + 3)2 + (y - 5)2 = 48. The coordinates of the center and the length of the radius of circle O are...

(1) Second time the circle equation is used in this exam. Flip the signs and simplify the square root of 48. (Not that you have to, there's only one possibility in the choices.)

13. In the diagram below of circle O, chord AB is parallel to chord CE.

A correct justification for mAC = mBD in circle O is

(1) Parallel chords intercept congruent arcs.

14. What is theslope o fa line perpendicular to the line whose equation is 3x - 7y + 14 = 0?

(2) Perpendicular again. The slope of the given line is 3/7. The slope is the inverse reciprocal, which is -7/3.

15. Line segment AB has endpoint A located at the origin. Line segment AB is longest when the coordinates of B are ...

(2) The question essentially asks, which of the four choices is the farthest from the origin. You might also notice that if you add the absolute value of x and the absolute value of y, in each case, the sum is 10. However, that isn't how you find distance. The distance formula involves squaring numbers, so it shouldn't be a surprise that (2, -8) is the farthest point.

16. In triangle FGH, m<F = m<H, GF = x + 40, HF = 3x - 20 and GH = 2x + 20. The length of GH is ...

(3) Because angles F and H are congruent, it is an isosceles triangle with legs GF = GH. So x + 40 = 2x + 20. (Note: HF does not matter in this.) Subtract x from both sides, and subtract 20 from both sides, and you find that x = 20. THAT IS NOT THE ANSWER. Plug 20 in for x, and 2(20) + 20 = 60.

17. In the diagram of quadrilateral ABCD, diagonals AEC and BED are perpendicular at E. [DIAGRAM SHOWS A KITE-SHAPE QUADRILATERAL WITH TWO DIAGONALS.]
Which statement is always true based on the given information?

(4) If the lines are perpendicular, then four right angles are formed, which are all 90 degrees, and therefore all congruent to each other.

18. Which set of numbers could represent the lengths of the sides of a right triangle?

(4) {8, 15, 17} is the only Pythagorean Triple listed. You should know it just from seeing it. If you didn't know it, you had to check each using the Pythagorean Theorem, although you should have realized that {7, 7, 12} couldn't be correct because the hypotenuse of that isosceles triangle would have to be 7(root 2).

19. In quadrilateral ABCD, the diagonals bisect its angles. If the diagonals are not congruent, quadrilateral ABCD must be a

(3) It's a rhombus, but not a square. I think I wrote that once before.

Which equation represents the line passing through P and parallel to line m?

(2) The line has a slope of 2, so the parallel line also has a slope of 2. The equation was given in point-slope form: y - y0 = m(x - x0).

21. Which compound statement is true?

(1) A disjunction (OR) only needs one part to be true. A square has four sides.

22. In triangle CAT, m<C = 65, m<A = 40 and B is a point on side CA, such that TB is perpendicular to CA. Which line segment is shortest?

(2) In any triangle, the side opposite the smallest angle is the smallest side, but there's a complication here. The triangle is cut into two smaller triangles, and because CAT is NOT a right triangle (it's 65-40-75), we can't apply the Right Triangle Altitude Theorem, even if that would have been of some use. Because TB is an altitude, it creates two right angles, so we can find the size of the smaller angles that T has been cut into, 50 and 25. The side opposite the 25-degree angle is going to be the smallest of the four segments listed.

23. In the diagram of triangle ABC below, DE || BC, AD = 3, DB = 2, and DE = 6.

What is the length of BC?

(2) Because the lines are parallel, the triangles are similar and the sides are proportional. 3 : 6 :: (3 + 2) : x, or 3x = 30. So x = 10.

24. In triangle ABC, an exterior angle at C measures 50 degrees. If m<A > 30, which inequality must be true?

(1) It must be less than 20 because of the Exterior Angle Theorem.

25. Which graph represents the graph of the equation (x - 1)2 + y2 = 4?

(2) Third time they are asking about the equation of a circle.

26. The equations of lines k, p, and m are given below:

k: x + 2y = 6
p: 6x + 3y = 12
m: -x + 2y = 10

Which statement is true?

(1) Lines p and m have perpendicular slopes (-2 and 1/2).

27. Peach Street and Cherry Street are parallel. Apple Street intersects them, as shown in the diagram below:

If M<1 = 2x + 36 and M<2 = 7x - 9, what is M<1?

(4) The two angles are supplementary, so 2x + 36 + 7x - 9 = 180. (They are NOT congruent, equal to each other.)
Therefore 9x + 27 = 180, 9x = 153, and x = 17. Again, that is NOT the answer.
Substitute 17 for x: 2(17) + 36 = 34 + 36 = 70 degrees.

28. A regular pyramid has a height of 12 centimeters and a square base. If the volume of the pyramid is 256 cubic centimeters, how many centimeters are in the length of one side of its base?

(1) 256 * 3 = 768. 768 / 12 = 64. The square root of 64 is 8.

That's it for the multiple-choice. I hope you did well.

As always, if there are any typos in the above, please holler at me at your earliest convenience so I can adjust things.

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