June 2022 Geometry Regents, Part 2

The following are some of the multiple questions from the recent January 2020 New York State Common Core Geometry Regents exam.

June 2022 Geometry, Part II

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

25. The Leaning Tower of Pisa in Italy is known for its slant, which occurred after its construction began. The angle of the slant is 86.03° from the ground. The low side of the tower reaches a height of 183.27 feet from the ground.

Determine and state the slant height, x, of the low side of the tower, to the nearest hundredth of afoot.

Answer:
First off, this isn't your typical Regents trigonometry ratio question, but it is still just a trigonometry question.

You have a right triangle, a leg (the height), a base angle that is opposite the leg, and you're looking for the hypotenuse. That means that you need to use sine.

sin 86.03 = 183.27 / x
x = 183.27 / sin 86.03
x = 183.7108...

To the nearest hundredth of a foot, the slant height is 183.71.

26. In the diagram below, quadrilateral ABCD is inscribed in circle 0, and mCD : mDA : mAB : mBC = 2:3:5:5. (refering to the arcs)

Determine and state m∠B.

Answer:
The sum of the four arcs is 360 degrees. Write an equation from the proportion:

2x + 3x + 5x + 5x = 360
15x = 360
x = 24

Since ∠B is an inscribed angle, it is half the size of the arc that it intercepts. It intercepts arc CDA. So ∠B = 1/2( 2(24) + 3(24)) = 60.

27. In the diagram below, a right circular cone has a diameter of 10 and a slant height of 13. Determine and state the volume of the cone, in terms of ℼ.

Answer:
The volume of a cone is 1/3 of the volume of a cylinder with the same base and height.

The radius is half of 10, which is 5. The height can be found using Pythagorean Theorem, with the radius being the other leg and the slant height being the hypotenuse of the right traingle.

I constantly remind students that you should know and recognize 3-4-5 an 5-12-13 triangles when you see them.

If you didn't, then 13² - 5² = h², so 169 - 25 = 144 = h², and h = 12.

V = 1/3 ℼ r² h = 1/3 ℼ (5)²(12) = 1/3 ℼ (25)(12) = 100ℼ

28. In the diagram below, parallelogram EFGH is mapped onto parallelogram IJKH after a reflection over line l.

Use the properties of rigid motions to explain why parallelogram EFGH is congruent to parallelogram IJKH.

Answer:
A reflection is a rigid motion where size (distance) and shape (angle measure) are preserved, so the image will be congruent to the original.

29. Izzy is making homemade clay pendants in the shape of a solid hemisphere, as modeled below. Each pendant has a radius of 2.8 cm.

How much clay, to the nearest cubic centimeter, does Izzy need to make 100 pendants?

Answer:
The volume of a hemisphere is 1/2 the volume of a sphere. So the volume is V = 1/2 (4/3 ℼ r³). You are given the radius, so there's no reason to divide it. Use the ℼ key on the calculator to get enough decimals to avoid a rounding error. DO NOT USE 3.14. (And forget about 22/7!)

V = 1/2 (4/3 ℼ r³)
V = 1/2 (4/3) (3.141592) (2.8)³
V = 45.976

The Volume for 100 pendants is 4598.

30. Determine and state the coordinates of the center and the length of the radius of the circle whose equation is x² + y² + 6x = 6y + 63.

Answer:
The equation must be put into the standard form for a circle. That means getting all the x and y terms onto the left side and then Completing the Squares.

x² + y² + 6x = 6y + 63

x² + y² + 6x - 6y = 63

x² + 6x + y² - 6y = 63

x² + 6x + 9 + y² - 6y + 9 = 63 + 9 + 9

(x + 3)² + (y - 3)² = 81

The center of the circle is (-3, 3) and the radius is √(81) = 9.

Remember to flip the signs and to take the square root of 81.

31. 1 Use a compass and straightedge to construct a line parallel to AB through point C, shown below. [Leave all construction marks.]

Answer:
To make a parallel line, draw line CA. Construct an angle congruent to CAB with C as its vertex.

End of Part II

How did you do?

Questions, comments and corrections welcome.

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