Wednesday, August 28, 2019

August 2019 Geometry Regents, Part 2

The following are some of the multiple questions from the recent August 2019 New York State Common Core Geometry Regents exam.

August 2019 Geometry, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. In parallelogram ABCD shown below, m∠DAC = 98° and m∠ACD = 36°.

What is the measure of angle B? Explain why.

The opposite angles in a parallelogram are congruent, so m∠B = m∠D. Also, ACD is a triangle, so the sum of its angles is 180 degrees.
180 - 98 - 36 = 46 degrees.
Angle D is 46 degrees, so angle B is 46 degrees.

26. An airplane took off at a constant angle of elevation. After the plane traveled for 25 miles, it reached an altitude of 5 miles, as modeled below.

To the nearest tenth of a degree, what was the angle of elevation?

The angle of elevation is measured from the ground. That makes the altitude of 5 miles the side opposite of the angle, and the 25 miles is the hypotenuse. This means you need to use sine.
Sin x = 5/25
x = sin-1 (1/5) = 11.53...
The angle of elevation is 11.5 degrees, to the nearest tenth.

27. On the set of axes below, Triangle ABC = Triangle DEF.

Describe a sequence of rigid motions that maps Triangle ABC onto Triangle DEF.

Some possibilities

A 180 degree rotation of ABC about the origin, followed by a translation of 4 spaces up.

A reflection of ABC across the y-axis, followed by a reflection across the line y = 2.

A reflection of ABC across the y-axis, followed by a reflection across the x-axis, followed by a translation 4 spaces up.

A 180 degree rotation of ABC about the point (0, 2).

You could also pick any point A, B, or C, and translate ABC so that it moves the point onto the corresponding point of DEF, and then rotate 180 degrees about that corresponding point. (You need to fill in the specifics -- the previous sentence by itself will not give you credit.)

28. The vertices of Triangle ABC have coordinates A(-2,-1), B(10,-1), and C(4,4). Determine and state the area of Triangle ABC. [The use of the set of axes below is optional.]

Since A and B both have a y-coordinate of -1, you know that AB is a horizontal line and you can use that as the base of your triangle. The height of the triangle is the vertical distance from -1 to 4. And the area of the triangle is 1/2 (base) (height).

b = 10 - (-2) = 12
h = 4 - (-1) = 5
A = 1/2(12)(5) = 30

Note that 72/360 is 1/5, which after a year of Geometry, you should probably recognize. One-fifth of ten squared is 1/5 of 100, which is 20.

29. Using the construction below, state the degree measure of ∠CAD. Explain why.

A reverse construction question! I like it!

Triangle ABC is an equilateral triangle because sides AC and BC are constructed to be the same length of AB.
Ray AD is constructed to be a bisector of angle CAB.
Since ABC is equilateral, m∠CAB = 60 degrees. Therefore, m∠CAD = 30 degrees, which is half of ∠CAB.

Note: You could have used a less wordy explanation than that, but I like to explain.

30. In the diagram below of circle K, secant PLKE and tangent PZ are drawn from external point P.

If m LZ = 56°, determine and state the degree measure of angle P.

K is the center of the circle, so LE is a diameter splitting the circle into two semicircles.
This means that arc EZL has a measure of 180 degrees, of which arc LZ is 56 degrees.
Therefore, arc EZ is 180 - 56 = 124 degrees.

The measure of angle P can be found by taking half of the difference between arc EZ and LZ:
1/2 (124 - 56) = 1/2 (68) = 34 degrees.

Alternatively, because arc LZ measures 56 degrees, then central angle LKZ has a measure of 56 degrees. If you draw radius KZ, it will be perpendicular to tangent PZ. That means that PKZ is a right triangle and you already know the size of two of its angles.
Angle P + 56 + 90 = 180, so Angle P = 34 degrees.

31. A large water basin is in the shape of a right cylinder. The inside of the basin has a diameter of 8 1/4 feet and a height of 3 feet. Determine and state, to the nearest cubic foot, the number of cubic feet of water that it will take to fill the basin to a level of 1/2 foot from the top.

You want to find the volume of the cylinder, but to a height of 2 1/2 feet instead of 3 feet. The radius is half of the diameter, which is (1/2)(8 1/4) = 4 1/8 Use the pi key on your calculator to get extra decimals for pi. Do NOT use 3.14 because it will not be accurate enough.
V = pi * r2 * h
V = 3.141592... * (4.125)2 * 2.5 = 133.64 = 134 cubic feet.

End of Part II

How did you do?

Questions, comments and corrections welcome.

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