(x, why?): Geometry Problems of the Day (Geometry Regents, January 2025 Part I)

This exam was adminstered in January 2025.

January 2025 Geometry Regents

Part I

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

17. A glass fish tank is designed to be placed on a stand in the corner of a room with perpendicular walls. The tank can be modeled using part of a cylinder, as shown below. The inner length of the fish tank along the wall is 22 inches, and the height of the tank is 27 inches.

How much water, to the nearest gallon, does the fish tank hold? [1 gal = 231 in³]

(1) 44
(2) 59
(3) 89
(4) 178

Answer: (1) 44

The sides are prependicular because it fits into the corner of a room. That means that there is a 90 degree angle and this tank is 1/4 of a complete cylinder.

So volume = 1/4 π r² h
V = 1/4 π (22)² (27)
V = 3267π
V = 10263.583 in³

10263.583 * 1 gal / 231 in³ = 44.43...

The correct answer is Choice (1).

18. Line m, whose equation is y = -2x + 8, is dilated by a scale factor of 1/2 centered at the origin. Which equation represents the image of line m?

(1) y = -x + 4
(2) y = -2x + 4
(3) y = -x + 8
(4) y = -2x + 8

Answer: (2) y = -2x + 4

The dilation will not affect the slope of the line so Choices (1) and (3) can be eliminated.

Since the dilation is centered on the origin, the y-intercept of the image will be half the distance of the original y-intercept. Therefore, the y-intercept of the image is +4. This is Choice (2).

19.In right triangle RST below, altitude SV is drawn to hypotenuse RT.

Which statement is always true?

(1) RT/ST = ST/VT
(2) VR/VT = VT/VS
(3) RV/SV = SV/RT
(4) TR/VR = VR/SR

Answer: (1) RT/ST = ST/VT

There are three right triangles in this image and all three of them are similar, which means that hte corresponding sides are proportional. Look for the proportion that lists corresponding sides.

In Choice (1), the hypotenuse and the long leg of the biggest triangle are compared to hypotenuse and the long leg of the right side triangle. This is the correct answer.

In Choice (2), the first ratio compares two sides that are not in the same triangle while the second ratio has two sides in the right triangle. Eliminate Choice (2).

In Choice (3), the first ratio contains the short leg and long leg of the left triangle and the second ratio has the short leg of the right triangle and the hypotenuse of the big triangle. Eliminate Choice (3).

In Choice (4), the hypotenuse of the big triangle is over a leg of a different triangle. Eliminate Choice (4).

20. What is the measure, in radians, of a central angle that intercepts an arc length of 12π cm in a circle with a diameter of 36 cm?

(1) π/6
(2) π/3
(3) 2π/3
(4) 3π/2

Answer: (3) 2π/3

The diameter is 36 cm, so the radius is 18 cm. Divide the arc length of 12π by 18 and you'll get 12π/18, which simplifies to 2π/3.

This is Choice (3).

21.A regular nonagon has a center point, P. What degree of rotation about point P will carry the nonagon onto itself?

(1) 60°
(2) 90°
(3) 180°
(4) 200°

Answer: (4) 200°

Every ninth of a full rotation will map a nonagon onto itself because a nongon has nine sides.

Because 360/9 = 40, the correct choice must be a multiple of 40 degrees. The only possible answer is Choice (4).

22. If two sides of a triangle have lengths of 2 and 7, the length of the third side could be

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

Answer: (2) 8

The sum of the lengths of the two shortest sides of a triangle must be greater than the length of the longest side. Another way to say it is that the sum of any two sides always must be greater than the length of the third side.

If two sides are 7 and 2, then the third side must be less than 7 + 2 or 9 and greater than 7 - 2 or 5.

The third side CANNOT be 4, 5, or 9. Only 8 is a possible side length for this triangle.

Note that 5 and 9 would create overlapping line segments, not triangles. If the smaller sides are 2 and 4, then they couldn't intersect because the endpoints are too far away from each other.

The correct answer is Choice (2).

23. The car tire shown in the photograph below has a diameter of 2 1/4 feet.

Approximately how many rotations will the tire make in one mile?

(1) 373
(2) 747
(3) 1328
(4) 2347

Answer: (2) 747

Find the circumference of the tire, in feet, and then divide 5280 feet by that number.

C = (3.141592...)(2.25) = 7.068582...

5280 / 7.068582 = 746.967355

It would take 747 rotations to travel one mile, which is Choice (2).

I want to make a airplane joke here...

24. In quadrilateral TOWN, OW ≅ TN and OT ≅ WN. Which additional information is sufficient to prove quadrilateral TOWN is a rhombus?

(1) ON ⊥ TW
(2) TO ⊥ OW
(3) OW || TN
(4) ON and TW bisect each other.

Answer: (1) ON ⊥ TW

The properties of a rhombus are that the opposite sides are parallel, the four sides are congruent, and the diagonals are perpendicular. The sides are not perpendicular unless the rhombus is also a square.

In Choice (1), ON and TW are diagonals, so you need them to be perpendiuclar. This is the correct answer.

In Choice (2), TO and OW are consecutive sides. They are only perpendicular in squares, but do not have to be perpendicular otherwise. Eliminate Choice (2).

In Choice (3), OW and TN must be parallel but this is not sufficient to prove the shape is a rhombus. It could be a trapezoid. Eliminate Choice (3).

In Choice (4), the diagonals of a rhombus only bisest each other if the shape is also a square (because this is a property of rectangles, not rhombuses). Eliminate Choice (4).

End of Part I. Comments and questions welcome.

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