Wednesday, April 24, 2024

Geometry Problems of the Day (Geometry Regents, January 2024 Part I)

This exam was adminstered in January 2024.

More Regents problems.

January 2024 Geometry Regents

Part I

Each correct answer will receive 2 credits. No partial credit.

9. In the diagram below, △GHJ is dilated by a scale factor of 1/2 centered at point B to map onto △CDF.

If m∠DFC = 40°, what is m∠HJG?

(1) 20°
(2) 40°
(3) 60°
(4) 80°

Answer: (2) 40°

A dilation preserves the shape of the original object, so the corresponding angles are congruent. Since ∠DFC and ∠HJG are corrsponding, they are congruent.

The correct answer is Choice (2).

10. Directed line segment AJ has endpoints whose coordinates are A(5,7) and J(-10,-8). Point E is on AJ such that AE:EJ is 2:3. What are the coordinates of point E?

(1) (1,-1)
(2) (-5,-3)
(3) (-4,-2)
(4) (-1,1)

Answer: (4) (-1,1)

AE is 2/5th of the segment AJ and EJ is 3/5th of the segment.

The change is x coordinates is -15. Multiply (-15)(2/5) and you get -6. Subtract 6 from 5, and the x-coordinate is -1. Choice (4) is the only possible answer.

The change in y coordinates is also -6. Subtract 6 from 7 and you get 1 as the y-coordinate of E. E is (-1,1), which is Choice (4).

11. A tipping platform is a ramp used to unload trucks, as shown in the diagram below.

The truck is on a 75-foot-long ramp. The ramp is tipped at an angle of 30°. What is the height of the upper end of the ramp, x, to the nearest tenth of a foot?

(1) 68.7
(2) 65.0
(3) 43.3
(4) 37.5

Answer: (4) 37.5

The ramp is the hypotenuse and the height is opposite the angle. That means you need to use the sine ratio.

Sin 30 = x / 75

x = 75 sin 30 = 37.5, which is Choice (4).

The sine of 30 degrees is .5, so it is half the size of the rampl.

12. In the diagram below of right triangle MET, altitude ES is drawn to hypotenuse MT.

Roofing shingles are sold in bundles. Each bundle covers 33 1/3 square feet. What is the minimum number of bundles that must be purchased to completely cover both rectangular sides of the roof?
If ME = 6 and SM = 4, what is MT?

(1) 9
(2) 8
(3) 5
(4) 4

Answer: (1) 9

Right triangles with altitudes drawn have three similar right triangles which have proportional sides.

So SM/ME = ME/MT and 4/6 = 6/MT

Cross-multiply and 4 MT = 36 and MT = 9, which is Choice (1).

Always reread the question: ST = 5, but that isn't what they asked.

13. In the diagram below of square CASH, diagonals AH and CS intersect at Z.

Which statement is true?

(1) m∠ACZ > m∠ZCH
(2) m∠ACZ < m∠ASZ
(3) m∠AZC = m∠SHC
(4) m∠AZC = m∠ZCH

Answer: (3) m∠AZC = m∠SHC

The four corner angles of a square are right angles, which are 90 degrees. The diagonals of a square are perpendicular and meet at right angles, which are 90 degrees. The diagonals bisect the corner right angles, creating 45 degree angles. So every angle in the image is either 45 degrees or 90 degrees.

In Choice (1), both angles are 45 degrees. They are equal so eliminate Choice (1).

In Choice (2), both angles are 45 degrees. They are equal so eliminate Choice (2).

In Choice (3), both angles are 90 degrees. Choice (3) is the correct choice.

In Choice (4), angle AZC is 90 degrees but ZHC is 45 degrees. They are not equal so eliminate Choice (4).

14. In the diagram below of circle O, secants CFD and CHE are drawn from external point C.

If m DE = 136° and m∠C = 44°, then m FH is

(1) 46°
(2) 48°
(3) 68°
(4) 88°

Answer: (2) 48°

The measure of angle C is equal to one-half the difference of the two arcs it intercepts.

That is, C = 1/2 (DE - FH), so

44 = 1/2 (136 - FH)

88 = 136 - FH

FH = 136 - 88 = 48 degrees

The answer is Choice (2).

15. A right circular cylinder has a diameter of 8 inches and a height of 12 inches. Which two-dimensional figure shows a cross section that is perpendicular to the base and passes through the center of the base?

Answer: (4) [See Image]

A cross section that is perpendicular to the base of a cylinder will have the shape of a rectangle.

Since is goes through the center of the base, it width of the rectangle will be the diameter of the cylinder. The length of the rectangle will be the height of the cylinder. The only possible choice is Choice (4).

If the cross section was parallel to the base, then Choice (1) would have been the correct choice because a radius of 4 means a diameter of 8.

16. On the set of axes below, AB is drawn and passes through A(-2,6) and B(4,0).

If CD is the image of AB after a dilation with a scale factor of 1/2 centered at the origin, which equation represents CD?

(1) y = -x + 4
(2) y = -x + 2
(3) y = -1/2x + 4
(4) y = -/2x + 2

Answer: (2) y = -x + 2

A dilation of a line will create an image that is parallel to the original line (or it will be the same as the original line if the center of dialation is on the line.)

Since it is parallel, the slope will not change. The slope of the given line is -1, which can be seen from the rise and the run. Eliminate Choices (3) and (4).

The original line has a y-intercept of +4, which is 4 units away from the origin. The image would be 1/2 as far away from the origin, which means that the y-intercept of the image will be at +2. This is Choice (2).

More to come. Comments and questions welcome.

More Regents problems.

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