January 2014 Geometry Regents Discussion Thread, Part 2: The Open-Ended Questions

Link back to Part 1: Intro

It was easier to start a second thread than to edit the previous one. Actually, it wasn't -- I just decided to do this.

Unless you scored better than 75% of the multiple-choice questions correct, you need some points from the open-ended sections -- Parts II, III and IV -- to put you over the top. (If you did manage 22 multiple-choice questions correct, hey, congrats! Good for you!) I'm starting here because I'll be grading oodles of these tomorrow morning.

29. The diameter of a sphere is 5 inches. Determine and state the surface area of the sphere to the nearest hundredth of a square inch.

This is extremely easy, considering that they give you the formula on the reference table! Remember to use 2.5 as the radius, not 5. Use your calculator for pi. DO NOT use 3.14. It won't give you enough accuracy.

SA = (4)(3.141592)(2.5)^2 = 78.5398... = 78.54 sq in.

30. Using a compass and straightedge, construct the perpendicular bisector of AB.

Hint: Look at question 2. Any of those look familiar? Maybe see a good starting point?
Sorry, I'm not making the diagrams.

31. The endpoints of AB are A(3, -4) and B(7, 2). Determine and state the length of AB in simplest radical form.

Digression: (Or should I say "Tangent"?)
Everyone in my class got a question just like this one correct in practice! And everyone showed All The Steps!
And I'm sure that you didn't all copy from each other, with the first person copying the answer from the Internet!
Ri-i-i-i-i-ight?

(I'll use the abbreviation "sqrt" for the square root function.) The square root of 4-squared plus 6-squared = sqrt(16+36) = sqrt(52) = sqrt(4*13) = 2*sqrt(13)

32. A right prism has a square base with an area of 12 square meters. the volume of the prism is 84 cubic meters. Determine and state the height of the prism, in meters.

Did you look up the formula in the back of the book? Why? Okay, if you did, did you find it? If you said, "Yes", you probably got this WRONG. It's a basic formula, so it won't be there.

The volume of a prism (84 m³) is the Area of the Base (12m²) times the height. Divide 84/12 and get 7 meters.

34. Without repeating the equations, you can see by looking at them (no manipulation required) that one has a slope of positive one-half and the other has a slope of negative one-half. They are NOT inverse reciprocals -- i.e., the slopes don't have a product of -1 -- and they (obviously) are not the same, so, therefore, they are neither parallel nor perpendicular.

"YES!", you have to give an explanation. Your answer must be backed up with a reason or definition and that reason has to have some kind of work or evidence that your reason is true. Even for something this obvious.

I will be shocked -- SHOCKED -- to find that "neither" with nothing supporting it would be worth a point. Don't count on it.

34. Another locus question. (There was one in the multiple-choice questions.) Two meters from the row of corn is two vertical lines, one on either side of line c. Five meters from the scarecrow is a circle with a radius of 5, centered on point T. Important: Six point T is 6 meters from line c, then T is only 4 meters from the locus line on the left side of c. The circle will intersect that line in two places, but it will NOT cross c. (And the other vertical line is right out.) There should be two X's on the paper.

Part III: Four credit questions

35. I can't copy the picture. (It isn't online yet.) So let me describe it.
Triangle ABC is isosceles with AB = BC. CA is extended through A to point D, and DB is drawn.
The measure of: <D = x, <DAB = 5x - 30, <DBA = 3x - 60, AB = 6y - 8, BC = 4y - 2.
Only algebraic answers will receive full credit.

Find m<D: Find x where x + 5x - 30 + 3x - 60 = 180
9x - 90 = 180,
9x = 30
x = 30, m<D = 30 degrees.

Find m<BAC, which is supplementary to <DAB, which is 5(30) - 30 = 120 degrees.
So BAC = 180 - 120 = 60 degrees.

Find the length of BC: Solve 6y - 8 = 4y - 2
2y = 6
y = 3, so BC = 4(3) - 2 = 12 - 2 = 10

Find the length of DC: SAY WHAT? How are we supposed to figure that out? Is A the midpoint of DC? Do we know AC? Can we get AC?
No to all of that.
Use what you already know -- and if you made a mistake, GUESS.
I say that only because this problem makes sense only if BAD is a right triangle. Two of the angles are 30 and 60, so angle DBC is 90 degrees.
You can't use Pythagorean Theorem because we don't know the length of BD and can't figure it out. BUT... we do know that in a 30-60-90 right triangle, the hypotenuse is twice the size of the shorter leg, and BC is the shorter leg. And BC, we just found, has a length of 10. Therefore, DC is 20.

EDIT: On further review, the smaller triangle is equilateral and the triangle on the left is isosceles, so you can show that those four smaller line segments all have a length of 10, therefore DC is 20. I would still require some work to get this, but I've been informed that no work is necessary because you "can assume" that they did the work because how else would they know. Okay, it wasn't quite stated in that fashion, but it might as well have been.

36. When doing a Composition of Transformations (two transformations) you have to go RIGHT TO LEFT. If you do it the other way, that is a Conceptual Error and you will lose 2 points right off the top. You have to find the Translation OF THE Reflection, so the reflection gets done first.

Flip the triangle over the y-axis: the x-co-ordinates change signs, but the y-co-ordinates will remain the same.
Next Translate x co-ordinates +4 (to the right) and y-co-ordinates -5 (down) to get your final answer.

37. Right Triangle Altitude Theorem. AD/DC=DC/DB, and DC is 6. Additionally, AD:AB = 1:5.

Let AD be x. That makes AB = 5x, so DB = 4x.
So x/6 = 6/4x.
Cross-multiply: 4x² = 36, x² = 9, so x = 3.
AD = 3, DB = 4x = 4(3) = 12.
(Algebraic solution.)

37. A very strange proof. They gave you the Statements. You needed to supply (some of) the Reasons. You have to prove a Tangent-Secant rule: (RS)² = RA * RT.

Basically, St is perpendicular to RS because tangent lines are perpendicular to the radius (and therefore the diameter) and the point of tangency. Angle RAS is a right angle because it is an inscribed angle for a semicircle, so it is half of 180 degrees, which is 90 degrees. RST is congruent to RAS because all right angles are congruent. Triangles RAS and RST are similar because of the AA Theorem. Because they are similar, their corresponding sides are proportional. Since they are proportional, the product of the means is equal to the product of the extremes, which gives us the equation which we were looking for.

Questions? Comments? Corrections? (Surely, I'll get those! And I'll call you "Surely" if I feel like it!) Debate?

Multiple-choice as I get to them in the next thread.