(C)Copyright 2018, C. Burke.
He doesn't really use that window pole for much else. Too bulky for drawing straight lines.
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He doesn't really use that window pole for much else. Too bulky for drawing straight lines.
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Getting this comic put together today took a little luck of its own!
Happy St. Patrick's Day!
Funny thing, when I graphed this, I thought I could get it shaded by using < instead of "=", and then I couldn't understand when two of the petals went away ... Sigh.
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They're going to occu-pie the Teacher Center.
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Heliosphere is a small con, but growing. It's not one of the big flashy events with all the media guests. It has writers and editors in attendance, and they'll happily offer advice along with telling you of their latest projects.
Unlike the inaugural outing last year, I was not a panelist or program participant this time around. I was a plain old fan, with no commitments, free to go where I wanted. (That also make this review a tad more independent, I guess.) And there was pretty to do.
For the science fan, there were panels devoted to the mechanics of sci-fi, including a Sunday morning discussion on Quantum Mechanics, and their applications in real life.
But the big draw would be for the History buffs (and the Alternate History buffs), because the con hosted a 1632 Mini-con, based on the works of, and the world created, by Guest of Honor. In this alternate timeline, a piece of land that included the fictional town of Grantville, West Virginia, was transported in time and space to Germany in 1632, in the middle of the Thirty Years War. The residents had to adapt to their new home and survive hostile encounters. Their "future" tech is helpful to a point, but they have to start an industrial revolution of their own even as they form their own United States over a century early.
A fun panel on Friday consisted of the "Weird Tech" that they could create based on the knowledge they brought with them and the raw materials on hand.
The Gaming room was a good place to pass some time, although I didn't play too much. Personally, I don't want to start a game that'll pull me in for a couple of hours when there are other things going on. Card games and word games usually work best for me -- but those can fool you, too, so be wary!
Another highlight is the popular Books & Brews panels, where the "brew" is coffee. I had signed up in advance to sit in with a group with another Guest of Honor, Dr. Charles E. Gannon, author of the Caine Riordan series of novels, as well as some entries in the 1632 series. Rather than took about his own work, Gannon quite eagerly chose to speak to the attendees about their writing, as nearly everyone at the table had done some kind of writing, or was at least trying. He sympathized with my comment that most of my writing credits happened in a different century.
What made this a highlight was running into Dr. Gannon again, later in the evening, at one of the parties. He came up to me, and asked me about my writing, and where I wanted it to go. If he hadn't had a fan before, well, he sealed the deal here. The guy's for real. (And now I have to make sure I have something written and submitted -- and accepted?? -- if I encounter him again next year.)
I've already registered for next year, April 5-7, 2019. Guests to be announced. More information can be found on their website: http://www.heliosphereny.org/
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Okay, so maybe I'm Disturbed.
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There's another routine somewhere in there about washing and drying the butter dish, too.
In my house, there's actually an extra step because we keep a spare, unfrozen stick of butter in the refrigerator, which goes in the dish, and then a frozen stick replaces the unfrozen stick.
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Just friends, Five.
11 is very popular. It can be a twin prime, a cousin prime, a sexy prime, and even an octupus prime if such a thing existed.
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The following are some of the multiple questions from the recent January 2018 New York State Geometry Regents exam.
The questions and answers to Part I can be found here.
Each correct answer is worth up to 2 credits. Partial credit is available. Work must be shown. Correct answers without work receive only 1 point.
25. Given: Parallelogram ABCD with diagonal AC drawn
Answer: You can give either a paragraph or two-column proof. However, when writing a paragraph, you still need to remember to have all the statements and reasons.
ABCD is a parallelogram; Given.
AB = CD, AD = BC; Opposite sides of a parallelogram are congruent.
AC = AC; Reflexive property
triangle ABC = triangle CDA; SSS
26. The diagram below shows circle 0 with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle 0. [Leave all construction marks.]
Answer: Strategy: if you draw a perpendicular bisector between A and B, you will get a vertical line through the center of the circle O. Where the line intersects the circle will be the other two vertices of the square. Use your straightedge to draw the square.
27. Given: Right triangle ABC with right angle at C
If sin A increases, does cos B increase or decrease? Explain why
Answer: Cos B will increase because the sin A = cos B. This is only a partially correct response. You need to back it up with a definition or explanation.
Sine is the ratio of opposite over hypotenuse (you can write this as a fraction.)
Cosine is the ratio of adjacent over hypotenuse.
The side that is opposite angle A is the same side that is adjacent to angle B.
Therefore sin A and cos B are the same ratio.
28. In the diagram below, the circle has a radius of 25 inches. The area of the unshaded sector is 500 pi in^{2}.
Answer: Find the area of the circle. Subtract 500pi from it. Compare the result to the original as a fraction. Multiply that fraction by 360 degrees.
A = (pi)(r)^{2}
A = (pi)(25)^{2}
A = 625pi
The shaded area is 625pi - 500pi = 125pi
The fraction of the circle that's shaded is (125pi)/625pi) = 125/625 = 1/5
1/5 (360) = 72 degrees.
29. A machinist creates a solid steel part for a wind turbine engine. The part has a volume of
1015 cubic centimeters. Steel can be purchased for $0.29 per kilogram, and has a density of 7.95 g/cm3.
If the machinist makes 500 of these parts, what is the cost of the steel, to the nearest dollar?
Answer: Find the mass of one part, and multiply it by 500 to get the mass of 500 parts in grams. Divide by 1000 to change it into kilograms. Then multiply the mass by $0.29 per kilogram.
D = m/V
7.95 = m/1015
m = 1015(7.95) = 8069.25
8069.25 * 500 = 4,034,625
4,034,625 / 1000 = 4,034.625
4,034.625 * 0.29 = 1,170.04125
$1,170 (to the nearest dollar)
Note: If you do all this work and forget to round to the nearest dollar, you will lose 1 of the 2 available points.
That stinks, but there's nothing you can do about that.
30. In the graph below, triangle ABC has coordinates A(-9,2), B(-6,-6), and C(-3, -2), and triangle RST has coordinates R(-2,9), S(5,6), and T(2,3).
Answer: They are not congruent. If they were congruent then you could map ABC onto RST with a series of rigid motions. If you reflect ABC over the line y = -x, A would map to R and C would map to T, but B would map to (6, 6), not to S(5, 6). So the triangles are not congruent.
You could also use the distance formula to show that the lengths of the sides of the triangles are not the same, but you still need to mention the properties of rigid motions to get full credit.
31. Bob places an 18-foot ladder 6 feet from the base of his house and leans it up against the side of his house. Find, to the nearest degree, the measure of the angle the bottom of the ladder makes with the ground.
Answer: Sketch a little picture, if it helps. The ladder is the hypotenuse, and the distance to the house is the adjacent side.
Use cos x = adj / hyp
cos x = 6 / 18
x = cos^{-1}(6/18) = 70.5287793655... = 71 degrees.
End of Part II
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Questions, comments and corrections welcome.
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The following are some of the multiple questions from the recent January 2018 New York State Geometry Regents exam.
Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.
1. In the diagram below, a sequence of rigid motions maps ABCD onto JKLM.
Answer: (1) 53°.
Because they were rigid motions, the corresponding angles are congruent. A corresponds to J, which is 82 degrees; B corresponds to K, which is 104 degrees. L is given as 121 degrees. The quadrilateral has a total of 360 degrees.
360 - (82 + 104 + 121) = 53
2. IParallelogram HAND is drawn below with diagonals HN and AD intersecting at S.
Answer: (2) AS = 1/2 AD .
The diagonals of a parallelogram bisect each other.
3. The graph below shows two congruent triangles, ABC and A'B'C'.
Answer: (4) a reflection over the line y = x.
A translation wouldn't change the orientation. If there were a rotation, each point would have more 1 quadrant (for 90 degrees) or 2 quadrants (for 180 degrees).
4. A man was parasailing above a lake at an angle of elevation of 32° from a boat, as modeled in the diagram below.
Answer: (1) 68.6.
The height is opposite the 32° angle, and the hypotenuse of 129.5 is given. Use the sine ration: sin = opp/hyp
sin 32 = x / 129.5
x = 129.5 (sin 32) = 68.6245447...
Be sure to have the calculator in DEGREE mode! If the calculator is set to radians, you will get an incorrect answer that might seem to be reasonable (71.4), but isn't one of the choices.
Speaking of "reasonable", because 32 degrees is close to 30 degrees, you could have estimated a multiple choice answer as follows: in a 30-60-90 degree triangle, the side opposite the 30 degree angle will ALWAYS be half of the hypotenuse. In that case the height would be 64.75. Since the angle is 32 degrees, it should be a little bigger than this number. Only 68.6 would be reasonable.
5. A right hexagonal prism is shown below. A two-dimensional cross section that is perpendicular to the base is taken from the prism.
Answer: (2) rectangle.
Parallel to the base would give a hexagon. Perpendicular to the base, whether it's "front to back", "side to side" or any other direction will yield a rectangle.
6. In the diagram below, AC has endpoints with coordinates A(-5,2) and C(4, -10).
Answer: (1) (-2,-2).
A ratio of 1:2 means that AB is 1/3 the length and BC is 2/3 the length.
If you look at the change in the x-coordinates, the distance from -5 to 4 is 9. (4 - (-5) = 9.)
One third of 9 is 3, so the change in the x-coordinate from A to B is +3.
-5 + 3 = -2, so the only choice is (-2, 2).
Checking the y-coordinate: (-10) - 2 = -12. (1/3)(-12) = -4.
2 + (-4) = -2, which is the y-coordinate of B.
7. An ice cream waffle cone can be modeled by a right circular cone with a base diameter of 6.6 centimeters and a volume of 54.45(pi) cubic centimeters. What is the number of centimeters in the height of the waffle cone?
Answer: (3) 15.
V = (1/3)(pi)r^{2}h
54.45(pi) = (1/3)(pi)(3.3)^{2}h
h = 54.45(pi) / ((1/3)(pi)(3.3)^{2}) = 15
You could solve the equation for h first, or you can plug in the numbers first and then solve. Note that you can divide both sides of the equation by pi to remove it from the equation entirely. Note that you were given a diameter of 6.6, which makes the radius 3.3.
8. The vertices of triangle PQR have coordinates P(2,3), Q(3,8), and R(7,3). Under which transformation of triangle PQR are distance and angle measure preserved?
Answer: (4) (x,y) -> (x + 2, y + 3).
Distance is not preserved if one of the coordinates is multiplied, which eliminates choices (1), (2), and (3).
Moreover, if the coordinates are not multiplied by the same scale factor, then angle measure will not be preserved either.
Choice (4) is a translation, which preserves distance and angle measure.
9. In triangle ABC shown below, side AC is extended to point D with m<DAB = (180 - 3x)°, m<B = (6x - 40)°, and m<C = (x + 20)°.
Answer: (3) 60°.
According to the Exterior Angle Theorem, the sum of the exterior angle equals the sum of the two remote angles.
Therefore,
10. Circle O is centered at the origin. In the diagram below, a quarter
of circle O is graphed.
Which three-dimensional figure is generated when the quarter
circle is continuously rotated about the y-axis?
Answer: (4) hemisphere.
If you reflect the quarter circle, you would get a semicircle in two dimensions. In three dimensions, that would be a hemisphere.
11. Rectangle A'B'C'D' is the image of rectangle ABCD after a dilation centered at point A by a scale factor of 2/3. Which statement is correct?
Answer: (1) Rectangle A'B'C'D' has a perimeter that is 2/3 the perimeter of
rectangle ABCD.
The perimeter would shrink. It would only be 2/3 of the original.
The area would be reduced to (2/3)^{2}, or 4/9, of the original area.
12. The equation of a circle is x^{2} + y^{2} - 6x + 2y = 6. What are the coordinates of the center and the length of the radius of the circle? (1) center (-3,1) and radius 4 (2) center (3, -1) and radius 4 (3) center (-3,1) and radius 16 (4) center (3, -1) and radius 16
Answer: (2) center (3, -1) and radius 4.
Rewrite the formula into standard form,
(x - h)^{2} + (y - k)^{2} = r^{2}, by grouping the variables and then completing the squares:
13. In the diagram of triangle ABC below, DE is parallel to AB, CD = 15, AD= 9, and AB= 40.
Answer: (3) 25.
CD / DE = CA / AB
15 / x = (15 + 9) / 40
(15 + 9)x = (15)(40)
24x = 600
x = 25
14. The line whose equation is 3x - 5y = 4 is dilated by a scale factor of 5/3 centered at the origin. Which statement is correct?
Answer: (1) The image of the line has the same slope as the pre-image but
a different y-intercept.
When dilating a line, the slope will not change. The new line will either be parallel to the original line, or coincident (i.e., the same line) if the center of the dilation is a point on the line.
If the center of the dilation is a point on the line, then the y-intercept would not change either, but that is not the case in this example. If the origin, (0, 0), were a point on the line, then 3(0) - 5(0) = 4, which is not true.
15. Which transformation would not carry a square onto itself?
Answer: (3) a 180° rotation about one of its vertices.
A 180° rotation about the center of the square would carry it onto itself.
16. In circle M below, diameter AC, chords AB and BC, and radius MB are drawn.
Answer: (4) mAB = (1/2)m<ACB .
The measure of arc AB would be TWICE the measure of angle ACB. In other words,
the measure of angle ACB would be half of arc AB.
ABC must be a right triangle, because angle B must be a right angle because it intercepts a semicircle.
ABM must be isosceles because both AM and BM are radii.
The measure of arc BC is equal to the measure of its central angle, angle BMC.
17. In the diagram below, XS and YR intersect at Z. Segments XY and RS are drawn perpendicular to YR to form triangles XYZ and SRZ.
Answer: (4) XY / SR = YZ / RZ.
We have enough information to show that the two triangles are similar, but not congruent. Eliminate choices (2) and (3).
Angles Y and R are right angles because they are perpendicular to YR, so they are congruent.
Angles XZY and RZS are congruent because they are vertical angles.
Therefore, XYZ is similar to RSZ by AA.
That means that the corresponding sides are proportional in length. Choice (4) is a correct proportion
18. As shown in the diagram below, ABC || EFG and BF = EF.
Answer: (2) 68.75°.
Because BF = EF, then BEF is an isosceles triangle.
So angle BEF is congruent to angle FBE because they are the base angles and BFE is the vertex angle.
Angle BFE is congruent to angle CBF because they are alternate interior angles.
Then CBF = BFE = 42.5 degrees.
So x + x + 42.5 = 180
2x = 137.5
x = 68.75°
19. A parallelogram must be a rhombus if its diagonals
Answer: (4) are perpendicular to each other .
Diagonals of a rectangle are also congruent, so choice (1) is incorrect.
Diagonals of a rectangle also bisect each other, so choice (2) is incorrect.
Diagonals of a rhombus do bisect each other, so choice (3) is incorrect.
20. What is an equation of a line which passes through (6,9) and is perpendicular to the line whose equation is 4x - 6y = 15?
Answer: (1) y - 9 = (-3/2)(x - 6).
The original line is in Standard Form. Find the slope of the line, using -A/B, or by rewriting it into point-slope or slope-intercept form.
A = 4, B = -6, so the slope is (-4)/(-6) = 2/3
The slope of a line perpendicular to this is -3/2, the negative reciprocal. This eliminates choices (2) and (4).
Point-slope form is y - y_{0} = m(x - x_{0}), so the solution is
y - 9 = (-3/2)(x - 6)
21. Quadrilateral ABCD is inscribed in circle 0, as shown below.
If m<A = 80°, m<B = 75°, m<C = (y + 30)°, and m<D = (x -10)°,
which statement is true?
Answer: (4) x = 115 and y = 70.
The sum of angles A and C is 180 degrees, and the sum of angles B and D is 180 degrees.
Angle C is 180 - 80 = 100 degrees, so y = 70.
Angle D is 180 - 75 = 105 degrees, do x = 115.
22. A regular pyramid has a square base. The perimeter of the base is 36 inches and the height of the pyramid is 15 inches. What is the volume of the pyramid in cubic inches?
Answer: (2) 405.
The perimeter of the square base is 36, so each side of the square is 9 (not 6). The area of the base is 9^{2} = 81.
The volume of the pyramid is (1/3) times the Area of the Base times the height:
V = (1/3)(81)(15) = 405
23. In the diagram below of triangle ABC, <ABC is a right angle, AC = 12,
AD = 8, and altitude BD is drawn.
What is the length of BC?
Answer: (2) 4 * SQRT(3).
Triangle BCD is similar to ABC because they both have right angles and they both have angle C (Reflexive Property).
So you can compare leg / hypotenuse = leg / hypotenuse
DC / BC = BC / AC
(12 - 8) / BC = BC / 12
BC^{2} = (12)(4) = 48
BC = SQRT(48) = SQRT(16 * 3) = 4 * SQRT(3)
24. In the diagram below, two concentric circles with center 0, and radii OC, OD, OGE, and ODF are drawn.
Answer: (3) The length of arc EF is 1.5 times the length of arc CD.
OE is 1.5 times the length of OC because 6/4 = 1.5.
The outer circle is a dilation of the inner circle with a scale of 1.5 centered on O. That means that the lengths of the corresponding arcs will have a scale of 1.5 as well.
End of Part I.
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Yeah, try representing a 3D activity in a 2D comic with an extra dimension for humor, which is often on an imaginary axis ... Maybe it would've looked better if I hadn't insisted that the lines be functions.
Parallel lines always have the same slope and are always the same distance apart.
With curve lines, the distance bewteen them isn't constant -- however, the vertical distance for any x value is always constant. At least, in this example because it's a simple transformation of the function; i.e. B(x) = R(x) + 2.
What about other examples? Well, that's left as an exercise to the reader. Yeah, that's the ticket!
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Got stuck trying to do something with 'curling'.
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Not that Kitty is not a Rigid Motion because Kitties will change size and shape to fill the available space on your lap.
Seriously, three months ago I made a notation in my notepad that read
"(X, why?) explains transformations: reflection, translation, rotations, kitty."
without any notation what "kitty" was supposed to be.
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