Wednesday, January 28, 2015

January 2015 Geometry Regents exam, Multiple-Choice

Once again, here are the multiple-choice problems with explanations for today's New York State Geometry Regents exam. The open-ended problems will (most likely) appear tomorrow. Since I typed this up as quickly as I could, there are no images, graphs or diagrams included. I'll edit them as I am able to.

A common theme: there were several questions involving the equation of a circle, as there always are, as well as several questions which not only involved the Pythagorean Theorem, but very specifically a 3-4-5 triangle or a multiple of it.

I hope everyone did well. Part I was, in my opinion, very "get-able" -- not very difficult for a Regents exam.

As always, apologies in advance for typos. I did rush this a bit. Here is Part I.

1.  What is the solution of the system of equations graphed below? (Image omitted)

y = 2x + 1

y = x2 + 2x – 3

The solutions of a system are the points where the two lines intersect. In this problem, that would be at (-2, -3) and (2, 5). Choice (4).

If you look at the other choices, (0, -3) is the y-intercept of the parabola, (-1, -4) is the vertex and (-3,0) and (1, 0) are the roots of the parabola. None of those have anything to do with the linear equation.

 

2. What are the coordinates of the midpoint of the line segment with endpoints (2, -5) and (8, 3)?

 

Take the average of the x-coordinates and of the y-coordinates.  You get (5, -1) which is exactly in the middle. Alternatively, you could have used the scratch graph paper and plotted it instead of calculating.

 

3.  As shown in the diagram below, when hexagon ABCDEF is reflected over line m, the image is hexagon A’B’C’D’E’F’. Under this transformation, which property is not preserved? (Image omitted)

 

Orientation is not preserved. The object has the same shape and size, but the direction its facing switches.

 

4.  In the diagram of triangle ABC below, BD is drawn to side AC. (Image omitted)

If m<A = 35, m<ABD = 25, and m<C = 60, which type of triangle is BCD?

 

BCD is equilateral. Angles A and ABD add up to 60 degrees. This makes <ADB = 120 degrees. Angle BDC is supplementary to ADB, so it is 60 degrees. Angle C is 60 degrees as well, so <CBD must be 60 degrees. Triangle BCD has three 60-degree angles, so it is an equilateral triangle.

 

5. In the diagram below of rhombus ABCD, the diagonals AC and BD intersect at E. (Image omitted)

If AC = 18 and BD = 24, what is the length of one side of rhombus ABCD?

 

The diagonals of a rhombus are perpendicular and bisect each other. They form four congruent right triangles. Each side of the rhombus is the hypotenuse of one of those right triangles. AC = 18, so AE = 9. BD = 24, so BE = 12. If the two legs of a right triangle are 9 and 12, then the hypotenuse is 15. Note: You can use the Pythagorean Theorem, but you should know 9-12-15 because it’s a multiple of 3-4-5, which is the most common right triangle used in these problems! The answer is 15.

 

6. What are the truth values of the statement “Opposites angles of a trapezoid are always congruent.” and its negation?

 

The statement is false, which makes the negation (“Opposite angles of a trapezoid are not always congruent.” ) true.

 

7. What is the length of a line segment whose endpoints have coordinates (5, 3) and (1, 6)?

Use the distance formula, or sketch the line segment, make a right triangle and use Pythagorean Theorem. Either way, you will get the square root of (32 + 42), which is the square of (9 + 16), which is 5. Again, 3-4-5 is a common right triangle. Recognize it when you see it!

 

8.  In the diagram below of isosceles triangle ABC, the measures of vertex angle B is 80 degrees. If AC extends to point D, what is m<BCD? (Image omitted)

 

The vertex angle is 80 degrees, so the two base angles have a total of 100 degrees. The base angles are congruent, so each is 50 degrees.  Angle BCD is supplementary to a 50 degree angle, so it is 130 degrees.

 

9. A student used a compass and a straightedge to construct CE in triangle ABC as shown below. (Image omitted)

Which statement must always be true for this construction?

 

The construction shows the bisecting of angle C. So <ACE is congruent to < BCE.

 

10. In triangle, ABC, AB = 4, BC = 7, and AC = 10. Which statement is true?

 

The smallest angle is across from the smallest side and the biggest angle is across from the biggest side. (Triangle Inequality Theorem). The opposite angle will be the vertex that is NOT part of the line segment; i.e., <C is across from AB and is the smallest angle, and <B is the largest.  The correct choice is (2) m<B > m <A > m< C.  (Sorry about the formatting. It can’t be helped.)

 

11. A circle whose center has coordinates (-3, 4) passes through the origin. What is the equation of this circle?

 

Using Pythagorean Theorem, we can see that the radius of the circle is 5. Again, another 3-4-5 triangle!

Flip the signs and square the radius: (x + 3)2 + (y – 4)2 = 25

 

12. Point W is located in plane R. How many distinct lines passing through point W are perpendicular to plane R?

 

One.  Think of any vertical pole coming out of the ground. There’s only one possible vertical pole at that point in the ground.

 

13. In the diagram below (image omitted), line l is parallel to line m, and line w is a transversal.

If m<2 = 3x + 17 and m <3 = 5x – 21, what is m<1?

 

Angles 2 and 3 are supplementary. Angles 1 and 2 are congruent. Solve for x and then find the measure of angle 2 to get the answer.

 

3x + 17 + 5x – 21 = 180

8x – 4 = 180

8x = 184

x = 23

 

m<2 = 3(23) + 17 = 69 + 17 = 86. Choice (4).

 

Note: be careful. I subtracted 4 instead of adding first time through. My incorrect answer wasn’t one of the choices, so I knew to go back and check my work!

 

14. The diagram below is of circle O. (Image omitted – it has a circle with center O at (5, -3) and a radius of 4.)

Which equation represents circle )?

 

Again. Flip the signs and square the radius. (x – 5)2 + (y + 3)2 = 16.

 

15. In isosceles trapezoid QRST show below, QR and TS are bases. (Image omitted)

If m<Q = 5x + 3 and m<R = 7x – 15, what is m<Q?

 

Angles Q and R are congruent, so their measures are equal.

5x + 3 = 7x – 15

3 = 2x – 15

18 = 2x

9 = x

 

m<Q = 5(9)  + 3 = 45 + 3 = 48. Choice (2).

 

16. Triangle ABC is graphed on the set of axes below. (Image omitted)

What are the coordinates of the point of intersection of the medians of triangle ABC?

 

This one is easier than it sounds. The median from vertex A to the midpoint of BC is a horizontal line with a length of 6. The medians will meet at the centroid, which is 2/3 of the way from A to the midpoint. That means that it will be 4 units to the right of point A. The answer is (-1, 2).

 

If you drew the three medians (two would be enough, though), you could probably eyeball the correct answer from the choices given.  Only one really makes sense.

 

17. Given the statement, “If a number has exactly two factors, it is a prime number,” what is the contrapositive of this statement?

 

Negative both parts and flip it around. “If a number is not a prime number, it does not have exactly two factors.”  Choice (2).

 

18. Which graph represents a circle whose equation is (x – 2)2 + (y + 4)2 = 4. (Images omitted)

 

Third time! Flip the signs. The center is (2, -4). The radius is 2, which is the square root of 4. That makes Choice (3) the correct answer.

 

19. If two sides of a triangle have lengths of 4 and 10, the third side could be …

 

The third side must be greater than 6, which is 10-4, and less than 14, which is 10+4. The only choice that fits is Choice (1) 8.

 

20. The lines represented by the equations 4x + 6y = 6 and y = 2/3x – 1 are …

 

Put the first line into slope-intercept form:

4x + 6y = 6

6y = -4x + 6

y = -2/3x + 1

 

The lines have the different slope, so they are neither parallel nor the same line. Additionally, the slopes are not negative reciprocals, so they are not perpendicular. However, they will intersect.

 

21. In the diagram below of triangle ABC, DE || AB. (Image omitted)

If CD = 4, CA = 10, CE = x + 2 and EB = 4x – 7, what is the length of CE?

 

Set up a proportion. CD/DA = CE/EB.  Notice that I said DA, not CA. DA = 10-4 = 6.

 

So 4/6 = x+2/4x -7. Cross multiply add you get

4(4x – 7) = 6(x + 2)

16x – 28 = 6x + 12

10x = 40

X = 4

 

CE = 4 + 2 = 6. Choice (3).

 

22. Parallelogram ABCD with diagonals AC and BD intersecting at E is shown below (image omitted).

Which statement must be true?

 

Diagonals of a parallelogram bisect each other, but that does not appear in the choices.

Each diagonal is a transversal across two parallel lines, so the alternate interior angles are congruent. This is needed to prove choice (2).

 

23. In the diagram below of circle O, m<ABC = 24. (Image omitted)

What is m<AOC?

The inscribed angle is half the size of the central angle. If m<ABC is 24, then m<AOC = 48.

 

24. Triangle A’B’C’ is the image of triangle ABC after a dilation of 2. Which statement is true?

 

The original triangle will have sides that are half as long as the image. The angles will be the same. So the angle is choice (3), m<B = m<B’.

 

25. In the diagram of the circle below, AD || BC, arc AB = (5x + 30) degrees, and arc CD = (9x – 10) degrees. (Image omitted)

What is mAB?

 

If the chords are parallel, then the arcs they create are congruent.

9x – 10 = 5x + 30

4x = 40

X = 10

AB = 5(10) + 30 = 50 + 30 = 80. Choice (4).

 

26. The bases of a prism are right trapezoids, as shown in the diagram below. (Image omitted)

Which two edges do not lie in the same plane?

 

Choice (1), BC and WZ. You cannot find a piece of paper which lines up with both edges. They are skew.

 

27. In the diagram below, A’B’ is the image of AB under which single transformation? (Image omitted)

 

It’s flipped over the x-axis and slid back a few spaces. It’s a glide reflection.

 

28. For which diagram is the statement triangle ABC ~ triangle ADE not always true?

 

Choice (4), which shows a trapezoid, with A as the midpoint of the diagonals. Triangles ABC and ADE share no angles and have none which must be congruent. Therefore, they are not always similar.

 

Saturday, January 24, 2015

Reviews of the Last 3.14 Book I Read

The title is a little misleading as there's no way that I made it anywhere near a seventh of the way through one of the books. However, in keeping up with a goal to read more and to write more, I've uploaded four reviews to my reading blog, which started out as simply someplace for me to keep track of what I've read and of the plots and characters of the more obscure ones.

Of the four books, all four were e-books, and only one of them I've ever actually seen in print. Two were fantasy, one sci-fi (time travel), and the last falls into the self-help category.

The direct links are:

  • Shards of the Glass Slipper: Queen Cinder by Roy A. Mauritsen, 2012.

  • Dirty Machines by David Matthew Olson, 2014.

  • 365 Things I Learned the Hard Way (So You Don't Have To) by The Digital Writer (Jonathan Wondrusch), 2012.

  • The God in the Clear Rock by Lucian Randolph, 2011.

    So check 'em out. (Well, maybe not that last one.)

  • Friday, January 23, 2015

    Name That Line!

    (Click on the comic if you can't see the full image.)
    (C)Copyright 2015, C. Burke.

    I could have used the I to work in AIR and then JAW, but it was too much.

    I don't know if that's Wink Martindale, Jim Lange, Dennis James or one of the other oldtime emcees.





    Thursday, January 22, 2015

    (x, why?) Mini: Square

    (Click on the comic if you can't see the full image.)
    (C)Copyright 2015, C. Burke.

    Circle is envious because he'll never be as sharply dressed.





    Wednesday, January 21, 2015

    Pascals Triangle

    (Click on the comic if you can't see the full image.)
    (C)Copyright 2015, C. Burke.

    The 14C3 gifts of Christmas?

    This occurred to me a while ago, but after Christmas, and I decided that it couldn't wait until December. (And I'd likely forget about it.)

    While coming up with mathematical formulas and computer code for calculating this number, I overlooked a a very reliable reference tool: Pascal's Triangle.

    It has more uses than simply expanding polynomials because of its many properties.

    Some of these properties are as follows:

  • The "zeroth" element of each row is the number 1.
  • The first element of each row is the number of the row (keeping in mind that the top row is Row 0).
  • The second element of each row is are the consecutive triangle numbers, the sum of the consecutive numbers before it.
  • Which makes the third element of each row the sum of the consecutive triangle numbers.
  • And, finally, the position of each element in Pascal's Triangle corresponds to the number of Combinations designated by the notation nCr where n is the row and r is the element of that row.

    What this means is that for any given day in that song (The Twelve Days of Christmas):

  • nC1 refers to the day we're up to. On Day 7, 7C1 is 7.
  • n+1C2 refers to the total number of gifts given on the nth day. On Day 7, 8C2 is 28.
  • n+2C3 refers to the total number of gifts given altogether up to the nth day. On Day 7, 9C3 is 84.

    Applying this to the 12th day of the song:

  • 12C1 is 12.
  • 13C2 is 78.
  • 14C3 is 364.

    And I could've been finished a whole lot sooner. But I wouldn't have gotten a recursive comic out of that.

    One last thing: Did you ever try to make a poster of Pascal's Triangle? Have your students tried to do it for a math fair? The numbers start to get really big in the middle, really fast. (That's a post for another day.) But that's also why the poster in this comic is so large! Otherwise, it wouldn't be readable (and I had to modify the "364" so you could find it easily!).





  • Tuesday, January 20, 2015

    (x, why?) Mini: Octonions

    (Click on the comic if you can't see the full image.)
    (C)Copyright 2015, C. Burke.

    Now I'm wondering about octgarlic.

    Unfortunately, this is from a part of mathematics that has past me by. Something from after I started concentrating on my computer science studies in college.

    I've seen the word "octonion" before, but it just popped up when i was lazily checking through a list of math blogs I bookmarked a couple years ago but haven't really visited. Sadly, I don't always know at first glance what they're talking about, and I rarely have the time to do the "catch-up" work to figure it out.





    Friday, January 16, 2015

    Survey Says: Granularity!

    (Click on the comic if you can't see the full image.)
    (C)Copyright 2015, C. Burke.

    All I had for Annie was "Cumulative Frequencies Open, Captain.", but it fit neither the space nor the flow of the dialogue.

    True story. A very similar question appeared in an online survey, and that's what I imagined in my mind.

    Oh, and the History teacher would've been the doctor if I wanted to extend the metaphor, but it was going too far already.





    Thursday, January 15, 2015

    That Moment When . . . II

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    (C)Copyright 2015, C. Burke.

    At least most of the student work I'd hung up had been taken down. Most of it.

    Another teacher, new to the school this year, asked me about document cameras, and wondered why she had the only math classroom without one. Equipment moves around sometimes, and she could request one. She told me a few days later that she happened to open the bottom drawer of her desk and found the camera in that drawer.

    I stopped and stared in disbelief and told her the story. I was the first teacher to get one because the AP insisted I try it out. They were in such a hurry to set me up that they never locked it down or attached it to the desk or the computer. At the end of the year, after a brief though of larceny, I put it in that drawer so it would be out of sight and somewhat safe. Nearly a year and a half later, it was still in the same spot. Whoever had the room in the intervening time either hadn't used the drawer or didn't know how to use the camera.

    By the way, this was the very same camera which was used to lure me into my surprise 50th birthday party.





    Tuesday, January 13, 2015

    (x, why?) Mini: B for Bank?

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    (C)Copyright 2015, C. Burke.

    Alas, so what then is the buzz?

    (I bet you thought I was going to go with two bees or not two bees, didn't you? Not gonna do it.)





    Monday, January 12, 2015

    Write? Check!

    (Click on the comic if you can't see the full image.)
    (C)Copyright 2015, C. Burke.

    Check? Write!

    I still write a few, mostly for charitable organizations where it makes for a handy receipt.

    What I really need to write is a name for the Science teacher! He doesn't have one. Almost all of the other (human) characters have names -- even Judy and Chuck, who were only named in a comic that neither of them were in, but were named nonetheless!

    Maybe I should have a Contest. Or maybe a Suggestion Box. Does he look like a Tim? Gary? It shouldn't be anything too "out there" if only because Mike and Ken would've made fun of it by now (especially Ken). So what do you think?

    In the meantime, write "2015" on the first five or six blank checks in your board. You'll be glad you did.





    Friday, January 09, 2015

    (x, why?) Mini: Getting Around

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    (C)Copyright 2015, C. Burke.

    Around and around and around. That's a lot of pi.





    Thursday, January 08, 2015

    (x, why?) Mini: Midpoint Quadrilaterals

    (Click on the comic if you can't see the full image.)
    (C)Copyright 2015, C. Burke.

    It wasn't a typo. I just misread the title. I had hoped for a really cool article. Oh, well.

    Fun things happen if you connect the midpoints of the sides of a quadrilateral, forming a new, smaller quadrilateral, and then connect those midpoints, and then those . . .