## Tuesday, April 16, 2024

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

This exam was adminstered in January 2024.

More Regents problems.

### Part I

1. Which expression is equal to sin 30°?

(1) tan 30°
(2) sin 60°
(3) cos 60°
(4) cos 30°

The sin of x degrees is equal to the cos of (90 - x) degrees because what is opposite one angle is adjacent to the complementary angle.

In a 30-60-90 degree right triangle, the sine ratio for the 30 degree angle will use the same leg (and hypotenuse) as the cosine of the 60 degree angle.

2. In the diagram of △ SRA below, KP is drawn such that ∠ SKP ≅ ∠ SRA.

If SK = 10, SP = 8, and PA = 6, what is the length of KR, to the nearest tenth?

(1) 4.8
(2) 7.5
(3) 8.0
(4) 13.3

First of all, this question, as it appears on the NYS Eduction Regents website, IS WRONG. The two triangles are not and cannot be congruent unless they are literally the same triangle. They can however be SIMILAR which would have the symbol ~ instead of "≅". This response assumes similarity.

If the triangles are simmialr then the corresponding sides will be proportional.

SK / SP = KR / PA

10 / 8 = x / 6

8x = 60

x = 7.5, which is choice (2).

Since it is not stated that this is an isosceles trapezoid, the diagonals (not drawn) are not necessarily congruent. Eliminate Choice (2).

In a trapezoid, the bases do not have to be congruent. In fact, in standard HS Geometry, the bases cannot be congruent, because that would make it a parallelogram instead of a trapezoid. (When you get to advanced mathematics, parallelograms will become special cases of trapezoids the way that equilateral triangles are also isosceles.)

3.A rectangle is graphed on the set of axes below.

A reflection over which line would carry the rectangle onto itself?

(1) y = 2
(2) y = 10
(3) y = 1/2 x - 3
(4) y = -1/2 x + 7

This rectangle has two lines of symmetry: the horizontal and veritcal lines that split the rectangle in half.

The horizontal line is y = 2, which is Choice (1) and the vertical line is x = 10, which is not listed.

The other two lines are the equations for the diagonals of the rectangle. However, reflecting a rectangle over its diagonal will not carry the rectangle onto itself unless the rectangle is a square.

4. The surface of the roof of a house is modeled by two congruent rectangles with dimensions 40 feet by 16 feet, as shown below.

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?

(1) 20
(2) 2
(3) 39
(4) 4

How much area do the two rectangle comprise? Divide that by 33 1/3 and round up so that there is enough to finish the job.

The area is 2 * 40 * 16 = 1280.

1280 / 33 1/3 = 38.4, which rounds up to 39.

5. Which equation represents a line that is perpendicular to the line whose equation is y - 3x = 4?

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

Answer: (1) y = -1/3 x - 4

If they are perpendicular, then the slopes must be inverse reciprocals. Since all four choices have different slopes, that is all we need to know.

The line y - 3x = 4 can be rewritten as y = 3x + 4, which has a slope of 3.

Any line perpendicular to this line must have a slope of -1/3. The only choice is Choice (1).

6. A vertical mine shaft is modeled in the diagram below. At a point on the ground 50 feet from the top of the mine, a ventilation tunnel is dug at an angle of 47°.

What is the length of the tunnel, to the nearest foot?

(1) 47
(2) 54
(3) 68
(4) 73

Notice that they are looking for the tunnel (the hypotenuse) and not the length of the mine (the opoosite leg).

You need to use the cosine ratio to find the missing hypotenuse.

Cos 47 = 50 / x

So x = 50 / cos 47 = 73.3, which is about 73

7. On the set of axes below, △BLU has vertices with coordinates B(-3,-2), L(-2,5), and U(1,1).

What is the area of △BLU?

(1) 11
(2) 12.5
(3) 14
(4) 17.1

LU has a slope of -4/3 and UB has a slope of 3/4, which means that the two lines are perpendicular and that their lengths can be used as the base and altitude of the triangle.

Also if you use either Distance Formula or Pythagorean Theorem, you will find that the length of LU and UB are both 5. You could also have drawn right triangles and recognized 3-4-5 triangles as the most common right triangle that gets used in problems of this type all the time!

So A = 1/2 (5) (5) = 12.5, which is Choice (2).

8. OIn the diagram below, △CAR is mapped onto △BUS after a sequence of rigid motions.
If AR = 3x + 4, RC = 5x - 10, CA = 2x + 6, and SB = 4x - 4, what is the length of SB?

(1) 6
(2) 16
(3) 20
(4) 28

If CAR is mapped onto BUS, then SB corresponds to RC and is congruent to RC, which means their lengths are equal. So:

5x - 10 = 4x - 4

x = 6

If x = 6, then SB = 4(6) - 4 = 20, which is Choice (3).

More to come. Comments and questions welcome.

More Regents problems.

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