While I wait for the January 2018 Geometry Regents exams to become available, and for the go-ahead to publish the questions and answers, allow me to revisit the August 2017 exams, which I haven't gotten around to dealing with until now.

### August 2017, Geometry, Part I

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

**1.** *A two-dimensional cross section is taken of a three-dimensional
object. If this cross section is a triangle, what can not be the three-dimensional
object?
*

**(2) cylinder** You can't take a cross section of a cylinder and get a triangle as a result. Vertical cross sections of cones and pyramids are triangles. Slicing off the corner of a rectangular prism will give you a triangle.

**2.** *The image of triangle DEF is triangle D'E'F'. Under which transformation will
the triangles not be congruent?
*

**(4) a dilation with a scale factor of 3/2 centered at the origin ** A dilation will yield a similar triangle, but not a congruent one. However, choice (3) has a scale factor of 1, which won't change the size. (It won't do anything.)

**3.** *The vertices of square RSTV have coordinates R(-1,5), S(-3,1), T(-7,3), and V(-5,7). What is the perimeter of RSTV?
*

**(3) 4 (sqrt (20)) ** Use the distance formula to find the length of one side and then multiply by four, because it's a square with four equal sides.

d = sqrt( (-3 - -1)^{2} + (1 - 5)^{2})

= sqrt ( (-2)^{2} + (-4)^{2})

= sqrt ( 4 + 16 ) = sqrt (20), the length of ONE side of the square.

**4.** *In the diagram below of circle O, chord CD is parallel to diameter AOB and mCD = 130.
*

What is mAC?

What is mAC?

**(1) 25 **

The semicircle ACDB is 180 degrees. Arc CD is 130. 180 - 130 = 50

The sum of arcs AC and DB = 50, but because the chord is parallel to the diameter, the two arcs are equal.

Therefore arc AC = 50 / 2 = 25.

**5.** *Iin the diagram below, AD intersects BE at C, and AB || DE.
*

If CD = 6.6 cm, D# = 3.4 cm, CE = 4.2 cm, and BC = 5.25 cm, what is the length of AC, to the nearest hundredth of a centimeter?

If CD = 6.6 cm, D# = 3.4 cm, CE = 4.2 cm, and BC = 5.25 cm, what is the length of AC, to the nearest hundredth of a centimeter?

**(4) 8.25 **

Because of the vertical angles and the alternate interior angles, we know that the triangles are similar. That means that the corresponding sides are proportional.

CD / CE = AC / BC

6.6 / 4.2 = AC / 5.25

4.2 AC = (6.6)(5.25)

AC = (6.6)(5.25)/4.2 = 8.25

**6.** *As shown in the graph below, the quadrilateral is a rectangle.
*

Which transformation would not map the rectangle onto itself?

Which transformation would not map the rectangle onto itself?

**(3) a rotation of 180° about the origin **

Rotating 180 degrees about the origin would put the rectangle in quadrants II and III.

Rotating 180 degrees about the point (4, 0) would map it onto itself because (4, 0) is the center of the rectangle.

**7.** *In the diagram below, triangle ACD has points B and E on sides AC and AD, respectively, such that DE||CD, AB = 1, BC = 3.5, and AD 18.
*

What is the length of AE, to the nearest tenth?

What is the length of AE, to the nearest tenth?

**(4) 4.0 **

Since the lines are parallel, triangles ABE and ACD are similar and their sides are proportional.

AB / AC = AE / AD

1 / 4.5 = AE / 18

4.5 AE = 18

AE = 4.0

**8.** *In the diagram below of parallelogram ROCK, m∠C is 70° and m∠ROS is 65°.
*

What is m∠KSO?

What is m∠KSO?

**(4) 135° **

Angle R is congruent to Angle C, so angle R is 70.

Angle KSO is the exterior angle for triangle ROS. The remote angles are 65 and 70, so 65 + 70 = 135.

**9.** *In the diagram below, ∠GRS ∠ART, GR 36, SR 45, AR 15,
and RT 18.
*

Which triangle similarity statement is correct?

Which triangle similarity statement is correct?

** (4) Triangle GRS is not similar to triangle ART. **

45 * 15 = 675

36 * 18 = 648

The sides are not proportional (which should've obvious because 5*5=25 and 6*8=48).

**10.** *The line represented by the equation 4y = 3x + 7 is transformed by a dilation centered at the origin. Which linear equation could represent its image?
*

**(1) 3x - 4y = 9 **

A dilation would not change the slope. The slope of the original line is 3/4.

Choice (1) has a slope of 3/4.

**11.** *Given triangle ABC with m∠B = 62° and side AC extended to D, as shown below.
*

Which value of x makes AB = CB?

Which value of x makes AB = CB?

**(4) 121° **

For AB to be congruent to CB, angle A must be congruent to angle ACB.

So y + y + 62 = 180

2y + 62 = 180

2y = 118

y = 59

Using the Remote Angle Theorem, x = 62 + 59 = 121.

**13.** *A rectangle whose length and width are 10 and 6, respectively, is
shown below. The rectangle is continuously rotated around a straight line to form an object whose volume is 150π.
Which line could the rectangle be rotated around?
*

** (3) the vertical line of symmetry **

Which line is used will determine what the radius is and what the height is.

Quick check: if the height is 10, then r^{2} = 150 / 10 = 15, which isn't a perfect square.

if the height is 6, then r^{2} = 150 / 6 = 25, which is a perfect square, and the radius is 5.

Since the long side has a length of 10, then a radius of 5 means you need to use the vertical line of symmetry and not the short side.

**14.** *If ABCD is a parallelogram, which statement would prove that ABCD is a rhombus?
*

** (3) AC is perpendicular to BD **

What makes a parallel also a rhombus is that the consecutive sides are congruent. This isn't given. Choices (1) and (2) are also true for parallelograms that aren't rhombuses. (Rhombi?) However, another property of rhombuses is that the diagonals are perpendicular, which is not true for other parallelograms.

**15.** *To build a handicapped-access ramp, the building code states
that for every 1 inch of vertical rise in height, the ramp must extend out 12 inches horizontally, as shown in the diagram below.
*

What is the angle of inclination, x, of this ramp, to the nearest hundredth of a degree?

What is the angle of inclination, x, of this ramp, to the nearest hundredth of a degree?

** (1) 4.76 **

tan x = opp / adj

tan x = 1 / 12

x = tan^{-1}(1/12) = 4.763...

**More to come. Check back later ...**

**Continuing...**

**16.** *In the diagram below of triangle ABC, D, E, and F are the midpoints of
AB, BC, and CA, respectively.
*

What is the ratio of the area of triangle CFE to the area of triangle CAB?

What is the ratio of the area of triangle CFE to the area of triangle CAB?

**(4) 1:4
**

Because D, E, and F are midpoints, the DE, EF, and FD are all midsegments, which are half the length of the side of CAB that they are parallel to. This is enough information to show that the four smaller triangles are all congruent, which makes each one 1/4 of the total area. Thus, 1:4.

**17.** *The coordinates of the endpoints of are A(-8,-2) and B(16,6).
Point P is on AB. What are the coordinates of point P, such that AP:PB is 3:5?
*

**((1) (1,1) **

3 + 5 = 8, so point P is 3/8 of the distance from A to B.

change in x-coordinate: (16 - (-8)) * (3/8) = 24 * (3/8) = 9

change in y-coordinate: (6 - (-2)) * (3/8) = 8 * (3/8) = 3

P is at (-8 + 9, -2 + 3) = (1, 1).

**18.** *Kirstie is testing values that would make triangle KLM a right triangle
when LN is an altitude, and KM = 16, as shown below.
*

Which lengths would make triangle KLM a right triangle?

Which lengths would make triangle KLM a right triangle?

**(2)LM = 12 and NM = 9 **

Working through each of the choices:

Choice (1), LM = 13, KN = 6. This means that MN = 16 - 6 = 10 and LN = sqrt((6)(10)) = sqrt(60).

Also LN = sqrt((13)^{2} - (10)^{2}) = sqrt(169 - 100) = sqrt(69). Contradiction.

Choice (2), LM = 12 and NM = 9. This means KN = 7 and LN = sqrt ((9)(7)) = sqrt(63).

Also LN = sqrt((12)^{2} - (9)^{2}) = sqrt(144 - 81) = sqrt(63). **This is the answer.**

Choice (3), KL = 11 and KN - 7. This means MN = 9 and LN = sqrt ((9)(7)) = sqrt(63).

Also LN = sqrt((11)^{2} - (7)^{2}) = sqrt(121 - 49) = sqrt(72). Contradiction.

Choice (4), LN = 8 and NM = 10. This means KN = 6 and LN = sqrt ((10)(6)) = sqrt(60). Contradiction.

**19.** *In right triangle ABC, m∠A = 32°, m∠B = 90°, and AC = 6.2 cm.
What is the length of BC, to the nearest tenth of a centimeter?
*

**(1) 3.3 **

sin 32° = BC/AC

.5299 = BC / 6.2

BC = 3.285, approx 3.3

**20.** *The 2010 U.S. Census populations and population densities are
shown in the table below.
*

Based on the table above, which list has the states’ areas, in square miles, in order from largest to smallest?

Based on the table above, which list has the states’ areas, in square miles, in order from largest to smallest?

**(1) Illinois, Florida, New York, Pennsylvania**

Divide population by density to get each area

Forida: 18,801,310 / 350.6 = 53626 sq mi

Illinois: 12,830,632 / 231.1 = 55520 sq mi

New York: 19,378,102 / 411.2 = 47126 sq mie

Pennsylvania: 12,702,379 / 283.9 = 44742 sq mi

**21.** *In a right triangle, sin (40 - x)° = cos (3x)°. What is the value of x?
*

**(4) 25**

sin (~~O~~) = cos (90 - ~~O~~)

So 40 - x = 90 - 3x

and 2x = 50

Therefore x = 25

**22.** *A regular decagon is rotated n degrees about its center, carrying
the decagon onto itself. The value of n could be
*

**(4) 252°**

A decagon has 10 sides, so a rotation of (360 / 10) = 36 degrees would rotate the figure onto itself, as would any multiple of 36 degrees. Divide each choice by 36 to see which works. (Number Sense would tell you only (4) is possible)

**23.** *In a circle with a diameter of 32, the area of a sector is 512π/3. The
measure of the angle of the sector, in radians, is
*

**(2) 4π/3**

The radius of the circle is 16.

The area of a sector of a circle is (1/2)r^{2} ∅

(1/2)(16)^{2} ∅ = 512π/3

128 ∅ = 512π/3

∅ = 4π/3

If you forgot this formula, the area of the entire circle is 256π

(512π/3) / 256π = 2/3

2/3 of a circle = 2/3 * (2π) = 4π/3

**24.** *What is an equation of the perpendicular bisector of the line segment
shown in the diagram below?
*

**(4) 2y - x = 0 **

The line has a slope of -2, so the perpendicular slope is 1/2.

The midpoint of the line is (0, 0), which gives you the y-interept.

So y = 1/2 x + 0

y - 1/2 x = 0, or

2y - x = 0.

**End of Part I**

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