## Tuesday, September 17, 2013

### August 2013 Geometry Regents Discussion, Part 1

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You can look at the exam at the Regents website.

1. Given: Triangle ABD, BC is the perpendicular bisector of AD. Which cannot be proven?
The length of BC has nothing to do with AD, AC or CD. THe answer is (2).

2. In the diagram of circle O shown below, chord CD is parallel to diameter AOB and the measure of arc CD = 110. What is the measure of arc DB?
I swear that this is a repeat -- not just a similar problem. Maybe I've used it in class before, I don't know.
DB is half of the difference of 180 and 110, which is 35. Chord AC measures the other 35, for the full 180 degrees.

3. paraphrased What is the negation [of "One is a prime number."] and the truth value of the negation?
Just add "Not". Do not change the words. "One is not a prime number." This (the negation) is true.

4. Co-ordindates are given for a triangle, and then it is rotated 180 degrees. What kind of triangle is it?
The fact that a triangle has been rotated doesn't change what kind of triangle it is. The length of its sides and the size of its angles remain unchanged. There is NO NEED to find new co-ordinates. Use the original ones.
Eliminate "Isosceles" as a choice because every isosceles triangle has to be either acute, right or obtuse, which are the other three choices.
Looking at the points, you'll see that A and B have the same y-value, so it's a horizontal line. B and C have the same x-value, making it a vertical line. Therefore, AB is perpendicular to BC and ABC is a right triangle.

Sidenote: if it hadn't been "Right", if you couldn't find that two sides were perpendicular -- that is, had inverse reciprocal slopes -- how did they expect you to find if the triangle was either acute or obtuse? Were you supposed to find the lengths of the three sides and then determine if a^2 + b^2 < c^2, where a, b, and c are the three sides in order from smallest to biggest? If so, that's a lot of work for a multiple-choice problem.

5. What is an equation of the circle with center (-5,4) and a radius of 7?
The first of many questions about the equation of the graph of a circle: (x - h)^2 + (y - h)^2 = r^2
Plug in (-5) for h and 4 for k and 7 for r, you get (x + 5)^2 + (y - 4)^2 = 49.
If you forgot to flip the signs, you lost 2 points here and most likely 6 more points elsewhere because of one formula.

6. In triangle ABC, ∠A is congruent to ∠B, and ∠C is an obtuse angle. Which statement is true?
If angle C is obtuse, it is the biggest angle and the side across from it (AB) is the longest side of the triangle.

7. There is an illustration showing a triangle with two medians meeting at point F.
The length of AF is twice the size of FB.

8. In circle O, diameter AB intersects chord CD at E. If CE is congruent to ED, then ∠CEA is which type of angle?
The chord must be perpendicular to the diameter to be bisected by it.

9. paraphrase: three congruent triangles If ABC = JKL = RST, then BC must be congruent to ____?
A notation question. BC must be congruent to KL and ST. ST is listed as a choice.

10. A triangle with one side extended. The exterior angle and the two remote angles are labeled with algebraic expressions.

Solve for x using the equation (x + 40) + (3x + 10) = 6x, which gives you 4x + 50 = 6x, 2x = 50, x = 25. This is NOT the final answer.
To find the size of the angle, add 25 + 40, which is 65 derees.

11. The bases of a right triangular prism are triangles ABC and DEF. Angles A and D are right angles, AB = 6, AC = 8, and AD = 12. What is the length of edge BE?
Visualize the prism. This is NOT a Pythagorean Theorem problem. They are not looking for BC or EF. BE is the same height as AD, which is 12.

12. A second circle equation question. Center is (-4, 1), radius is 3. Use the formula from question 5.

13. The illustration shows corresponding angles are congruent.

14. The lateral area of a right circular cone is equal to 120(pi) cm^2. If the base of the cone has a diameter of 24 cm, what is the length of the slant height, in centimeters?
The reference table lists Lateral Area (L) for a Right Circular Cone as L = (pi)r l where l is the slant height
Half of 24 is 12, so 120(pi) = (12)r(pi), and r = (120*pi)/(12*pi) = 10.

15. Is the system of equations parallel, perpendicular, the same line or something else? Find the slope of each of the lines. You can rewrite the equations in slope-intercept form (y=mx+b) or in Standard form (Ax + By = C) and use m=-A/B.
3y + 6 = 2x has a slope of 2/3.
2y - 3x = 6 has a slope of 3/2.
Those are reciprocals, but they are not inverse (or negative) reciprocals. So they intersect, but not at right angles.

16. In a coordinate plane, the locus of points 5 units from the x-axis is the
First off, I have a problem with the use of the word "is", which indicates a singular answer, when two of the choices are plural, including the correct choice.
Five units above or below the x-axis (y=0) are the lines y = 5 or y = -5, respectively.

17. The sides of a triangle are 8, 12, and 15. The longest side of a similar triangle is 18. What is the ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle?
The ratio between corresponding sides of the smaller to the larger is 15/18 or 5/6, or 5:6. The perimeter is the sum of the sides and will have the same ratio.

18. What is the converse of the statement “If lines m and n are parallel, then lines m and n do not intersect”?
q -> p: If lines m and n do not intersect, then lines m and n are parallel.

19. When the system of equations y + 2 = (x - 4)^2 and 2x + y - 6 = 0 is solved graphically, the solution is?
You could work backward from the choices, or you could expand (x-4)^2 and solve the system of equations to get the answer. Neither (6, 6) nor (-2, 2) work in the linear equation, so they could be eliminated immediately. (4, -2) works in the linear equation and the parabola.

20. shortened: If VW = 7x - 3 and AB = 3x + 1, what is the length of VC?
AB is the midsegment of a triangle, VW is the parallel side, VC is half of VW and is congruent to AB.
Solve for x, using: 7x - 3 = 2(3x + 1), which yields x = 5, which again is NOT the final answer.
Substitute 5 for x in 3x + 1, and the answer is 16.

21. Two prisms have equal heights and equal volumes. The base of one is a pentagon and the base of the other is a square. If the area of the pentagonal base is 36 square inches, how many inches are in the length of each side of the square base?
Seriously?
Volume = Area of Base X height. The volumes are the same, and the heights are the same, so the areas of the bases are the same; i.e., 36.
Since it's a square, the length of a side is the square root of 36, or 6. (Do NOT divide by 4 and get 9, which is choice (2)!)

22. What is the difference between the sum of the measures of the interior angles of a regular pentagon and the sum of the measures of the exterior angles of a regular pentagon?
The interior adds up to 540 degrees, and the exterior to 360. That's a difference of 180.

23. If line ℓ is perpendicular to distinct planes P and Q, then planes P and Q _________ ... are parallel.

24. Another equation of a circle problem, this time with graphs. Hint: the radius is only 2. Second hint: the center is (0, 2), not (0, -2).

25. There's a circle with center B and tangents AC and AD. AB, BC and BD are drawn. AB = 15 (read the problem, not the picture!) and AC = 12.
It's a right triangle, so the radius of the circle (BC and BD) is 9. (9-12-15 is a multiple of 3-4-5. You should KNOW that. You shouldn't need to calculate it again.)

26. Triangle ABC is a right triangle with altitude AD drawn to the hypotenuse BC. If BD = 2 and DC = 10, what is the length of AB?
AB is the hypotenuse of the smaller triangle, and 2 is the shorter leg of the smaller triangle. BC, which is 12, is the hypotenuse of the largest triangle, and AB is the base of it. So BD:AB = AB:BC, or (AB)^2 = (BD)(BC)=(2)(12)=(24). So, (AB)=24^(.5), which is 2(6)^(.5).

Alternatively,
Solve this equation (AD)^2 = (BD)(DC) = (2)(10) = 20
You get the square root of 20.
Now (2)^2 + (20) = (AB)^2.
(AB)^2 = 24, again. Same answer.

27. Triangle ABC has vertices A(0,0), B(6,8), and C(8,4). Which equation represents the perpendicular bisector of BC?
Find the slope of BC: (4 - 8) / (8 - 6) = (-4)/2 = -2.
Any line perpendicular to BC must have a slope of positive (1/2).
Only one choice has that, so the fact that the line is a bisector is meaningless in this context.
Had it been open-ended, you would have had find the midpoint of BC and plug those co-ordinates into the y = (1/2)x + b and solved for b, the y-intercept. Doesn't sound like a whole lot of fun.

28. Chords AB and CD intersect at point E in a circle with center at O. If AE = 8, AB = 20, and DE = 16, what is the length of CE?
(AE)(EB) = (CE)(ED), so (8)(12) = (CE)(16)
CE = 6.
Notice that they gave you AB and not (EB).

End of Part I.

Most mistakes are likely typos that snuck in because I didn't proofread carefully and because I'm still not getting enough sleep (because I'm typing up things like this).

If you find mistakes that are typographical in nature, please point them out so I can adjust them.
If my math is wrong on a problem -- if I fell into one of those traps I warn my students about -- please, feel free to explain and give the correct solution.