January 2016 New York Geometry (Common Core) Part 2

It's that time of year again: Regents time. I will be uploading parts of the Regents exams for Geometry and Algebra over the next few days. Please, be patient -- I have to type all this in by hand. And that doesn't count the amount of stuff that I need to scan in.

As always, the images will be scanned in when available, and the entries will be updated.

New York Geometry (Common Core) Part 2

25. Triangle ABC is graphed on the set of axes below. (image omitted) Graph and label triangle A’B’C’, the image of triangle ABC after a reflection over the line x = 1.

There are two parts to this: you need to both graph it and label it to get both of them.
The line x = 1 is the vertical line one unit to the right of the y-axis. So your answer is a triangle with vertices A’ (5, 0), B’(2, 4) and C’(2,0).
The vertices need to be labeled correctly, or you could probably get away with writing the labels and coordinates below the grid.
If you forgot something, you lose a point. If you do some other transformation, including a reflection over a different line, you will lose half-credit, which is one point.

26. In the diagram below (image omitted) of circle O with diameter BC and radius OA, chord DC is parallel to chord BA.

If m <BCD = 30^o, determine and state m <AOB.

Because DC is parallel to BA with BC as a transversal, then the alternate interior angles are congruent. So <DCB = <ABC = 30 degrees.
Because OB and OA are both radii, OB = OA, so triangle OAB is an isosceles triangle, with base angles equal to 30 degrees. 30 + 30 = 60. 180 – 60 = 120. m <AOB = 120^o

If you solved this using inscribed angles and arcs, more power to you. If you got the correct answer, you will get full credit.

27. Directed line segment PT has endpoints whose coordinates are P(-2, 1) and T(4, 7). Determine the coordinates of point J that divides the segment in the ratio 2 to 1.
The use of the set of axes below is optional.

The difference of the x values is 6. The difference in the y values is also 6. Two thirds of 6 is 4.
Add 4 to the x and y value of P. J(-2 + 4, 1 + 4) = J(2, 5)

28. As graphed on the set of axes below (image omitted), triangle A’B’C’ is the image of triangle ABC after a sequence of transformations.

Is triangle A’B’C’ congruent to triangle ABC? Use the properties of rigid motions to explain your answer.

Triangle A’B’C’ is the image of ABC reflected over the y-axis and also translated down 3 units. Reflections and translations are rigid motions that do not affect the size or shape, so the image is congruent.

29. A carpenter leans an extension ladder against a house to reach the bottom of a window 30 feet above the ground. As shown in the diagram below, the ladder makes a 70^o angle with the ground. To the nearest foot, determine and state the length of the ladder.

This used to be an Algebra topic, but now is in Geometry, involving Trigonometry ratios.

The ladder and the wall form a right triangle. You know the bottom angle. You have the length of the opposite side. You need the length of the hypotenuse. Opposite and Hypotenuse means using Sine.

Your equation is sin(70) = 30/x
Multiply by x: x sin(70)= 30
Divide by sin(70): x = 30/sin(70)
Put that in your calculator and get x = 31.925 = 32 feet.

You will lose a point if you use the incorrect ratio, for making an incorrect calculation, or not rounding correctly. Zero if you make more than one mistake.

You will not be penalized twice for a consistent error.

30. During an experiment, the same type of bacteria is grown in two petri dishes. Petri dish A has a diameter of 51 mm and has approximately 40,000 bacteria after 1 hour. Petri dish B has a diameter of 75 mm and has approximately 72,000 bacteria after 1 hour.

(image omitted)

Determine and state which petri dish has the greater population density of bacteria at the end of the first hour.

The population density is the population divided by the Area. You need to find the area of each circle and then divide the population by that area.

Don’t forget to divide the diameters by 2 to get the radii.

A_A = (pi)(25.5)² = 2042.82…
Divide 40000 by 2042.82 = 19.58

A_B = (pi)(37.5)² = 4417.86…
Divide 72000 by 4417.86 = 16.297

Petri dish A has the greater population density.

Important: Normally, you shouldn’t round in the middle of a problem. I only did it here because a) I didn’t need an exact answer, and b) I have to divide by the number that I rounded. I could have written the equations to avoid that, but then I wouldn’t have found the individual areas. Not that they were needed in this problem.

31. Line L is mapped onto line M by a dilation centered at the origin with a scale factor of 2. The equation of line L is 3x – y = 4. Determine and state an equation for line M.

This is a surprisingly simple question to answer, but a little more complicated to explain. The slope of the line will not change. The x-intercept and y-intercepts will double (be twice as far away from the origin).

So the answer is obviously 3x – y = 8 or some variation.

But how do you show it?

You could rewrite the equations in slope intercept form, and then double the y-intercept: So 3x – y = 4 becomes 3x – 4 = y, with the answer of y = 3x – 8.

Or you can state that the slope doesn’t change, so the equation stays 3x – y = C
When x = 0, 3(0) – y = 4 when y = -4. Double -4 to -8
Evaluate 3(0) – (-8) = C, C = 8. So 3x – y = 8.

32. The aspect ratio (the ratio of screen width to height) of a rectangular flat-screen television I s16:9. The length of the diagonal of the screen is the television’s screen size. Determine and state, to the nearest inch, the screen size (diagonal) of this flat-screen television with a screen height of 20.6 inches.

This can be solved using ratios and Pythagorean Theorem, or by using Trigonometric Ratios.

First, set up a proportion 16/9 = x/20.6 and cross-multiply.
9x = 329.6, x = 36.6
20.6² + 36.6² = c²
424.36 + 1339.56 = c²
1763.02 = c²
C = 41.999 = 42 inches

Second, the ratio 16:9 represents the opposite over the adjacent, which is the tangent of the top angle of the set. So tan(y) = 16/9 and y = tan-1(16/9) = 60.64 degrees.

Using the adjacent and the hypotenuse and cosine, we get the following:
cos(60.64) = 20.6/x, so x = 9/cos(60.64) = 42.0155… = 42 inches.