NYSED High School Regents Geometry

Last updated 12 months ago

35 questions

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

4

Required

4

Required

4

Required

6

Library

NYSED High School Regents Geometry

Last updated 12 months ago

35 questions

Part I 

Answer all 24 questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For each statement or question, choose the word or expression that, of those given, best completes the statement or answers the question. 

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

Required

2

PART II
Answer all 7 questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 

Required

2

Write an equation of the line that is parallel to the line whose equation is 3y + 7 = 2x and passes through the point (2,6).

Required

2

Parallelogram ABCD is adjacent to rhombus DEFG, as shown below, and \overline{FC} intersects \overline{AGD } at H.

If m\angle B = 118^\circ and m\angle AHC = 138^\circ, determine and state m\angle GFH.

Required

2

As shown in the diagram below, secants \overrightarrow{PWR} and \overrightarrow{PTS} are drawn to circle O from external point P.

If m\angle RPS = 35^\circ and m\overset{\frown} {RS} = 121^\circ, determine and state m\overset{\frown}{WT}.

Required

2

On the set of axes below, \triangle ABC is graphed with coordinates A(-2,-1), B(3,-1), and C(-2,-4). Triangle QRS, the image of \triangle ABC, is graphed with coordinates Q(-5,2), R(-5,7), and S(-8,2).

Describe a sequence of transformations that would map △ABC onto △QRS.

Required

2

Given points A, B, and C, use a compass and straightedge to construct point D so that ABCD is a parallelogram.

[Leave all construction marks.]

Required

2

On the set of axes below, \triangle DEF has vertices at the coordinates D(1,-1), E(3,4), and F(4,2), and point G has coordinates (3,1). Owen claims the median from point E must pass through point G.

Is Owen correct? Explain why.

Required

2

A walking path at a local park is modeled on the grid below, where the length of each grid square is 10 feet. The town needs to submit paperwork to pave the walking path. Determine and state, to the nearest square foot, the area of the walking path.

PART III

Answer all 3 questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 

Required

4

A triangle has vertices A(-2, 4), B(6, 2), and C(1, -1).

Prove that \triangle ABC is an isosceles right triangle.
[The use of the set of axes below is optional.]

Required

4

Theresa has a rectangular pool 30 ft long, 15 ft wide, and 4 ft deep. Theresa fills her pool using city water at a rate of $3.95 per 100 gallons of water.

Nancy has a circular pool with a diameter of 24 ft and a depth of 4 ft. Nancy fills her pool with a water delivery service at a rate of $200 per 6000 gallons.

If Theresa and Nancy both fill their pools 6 inches from the top of the pool, determine and state who paid more to fill her pool. [1 ft3 water = 7.48 gallons]

Required

4

As modeled in the diagram below, an access ramp starts on flat ground and ends at the beginning of the top step. Each step is 6 inches tall and 8 inches deep.

If the angle of elevation of the ramp is 4.76^\circ, determine and state the length of the ramp, to the nearest tenth of a foot. 

Length of the ramp: _______ feet

Determine and state, to the nearest tenth of a foot, the horizontal distance, d, from the bottom of the stairs to the bottom of the ramp.

Horizontal distance: _______ feet

PART IV

Answer the question in this part. A correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for the question to determine your answer. Note that diagrams are not necessarily drawn to scale. For the question in this part, a correct numerical answer with no work shown will receive only 1 credit. 

Required

6

In the diagram of quadrilateral ABCD with diagonal \overline{AC} shown below, segments GH and EF are drawn, \overline{AE} \cong \overline{CG}, \overline{BE}\cong\overline{DG}, \overline{AH} \cong\overline{CF}, and \overline{AD}\cong\overline{CB}.

Prove: \overline{EF} \cong\overline{GH}

Part I 

Answer all 24 questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For each statement or question, choose the word or expression that, of those given, best completes the statement or answers the question. 

After a dilation with center (0,0), the image of \overline{DB} is \overline{D'B'}. If DB = 4.5 and D'B' = 18, the scale factor of this dilation is:

\frac{1}{5}

5

\frac{1}{4}

4

In the diagram below, \triangle ABC with sides of 13, 15, and 16, is mapped onto \triangle DEF after a clockwise rotation of 90° about point P.

If DE = 2x - 1, what is the value of x?

7

7.5

8

8.5

On the set of axes below, \triangle ABC has vertices at A(-2,0), B(2,-4), C(4,2), and \triangle DEF has vertices at D(4,0), E(-4,8), F(-8,-4).

Which sequence of transformations will map △ABC onto △DEF?

a dilation of △ABC by a scale factor of 2 centered at point A

a dilation of △ABC by a scale factor of \frac{1}{2} centered at point A

a dilation of △ABC by a scale factor of 2 centered at the origin, followed by a rotation of 180^\circ about the origin

a dilation of △ABC by a scale factor of \frac{1}{2} centered at the origin, followed by a rotation of 180^\circ about the origin

The figure below shows a rhombus with noncongruent diagonals.

Which transformation would not carry this rhombus onto itself?

a reflection over the shorter diagonal

a reflection over the longer diagonal

a clockwise rotation of 90^\circ about the intersection of the diagonals

a counterclockwise rotation of 180^\circ about the intersection of the diagonals

In the diagram below of circle O, points K, A, T, I, and E are on the circle, \triangle KAE and \triangle ITE are drawn, \overset{\frown}{KE}\cong\overset{\frown}{EI}, and \angle EKA \cong\angle EIT.

Which statement about \triangle KAE and \triangle ITE is always true?

They are neither congruent nor similar.

They are similar but not congruent.

They are right triangles.

They are congruent.

In right triangle ABC shown below, point D is on \overline{AB} and point E is on \overline{CB} such that \overline{AC} \parallel \overline{DE}.

If AB = 15, BC = 12, and EC = 7, what is the length of \overline{BD}?

8.75

6.25

5

4

In rhombus VENU, diagonals \overline{VN} and \overline{EU} intersect at S. If VN = 12 and EU = 16, what is the perimeter of the rhombus?

80

40

20

10

Given right triangle ABC with a right angle at C, m\angle B = 61^\circ. Given right triangle RST with a right angle at T, m\angle R = 29^\circ.

Which proportion in relation to \triangle ABC and \triangle RST is not correct?

\frac{AB}{RS} = \frac{RT}{AC}

\frac{BC}{ST} = \frac{AB}{RS}

\frac{BC}{ST} = \frac{AC}{RT}

\frac{AB}{AC} = \frac{RS}{RT}

A vendor is using an 8-ft by 8-ft tent for a craft fair. The legs of the tent are 9 ft tall and the top forms a square pyramid with a height of 3 ft.

What is the volume, in cubic feet, of space the tent occupies?

256

640

672

768

In the diagram below of right triangle KMI, altitude \overline{IG} is drawn to hypotenuse \overline{KM}.

If KG = 9 and IG = 12, the length of \overline{IM} is

15

16

20

25

Which three-dimensional figure will result when a rectangle 6 inches long and 5 inches wide is continuously rotated about the longer side?

a rectangular prism with a length of 6 inches, width of 6 inches, and height of 5 inches

a rectangular prism with a length of 6 inches, width of 5 inches, and height of 5 inches

a cylinder with a radius of 5 inches and a height of 6 inches

a cylinder with a radius of 6 inches and a height of 5 inches

Which statement about parallelograms is always true?

The diagonals are congruent.

The diagonals bisect each other.

The diagonals are perpendicular.

The diagonals bisect their respective angles.

From a point on the ground one-half mile from the base of a historic monument, the angle of elevation to its top is 11.87°. To the nearest foot, what is the height of the monument?

543

555

1086

1110

The area of a sector of a circle with a radius measuring 15 cm is 75\pi \, \text{cm}^2. What is the measure of the central angle that forms the sector?

72°

120°

144°

180°

Point M divides \overline{AB} so that \text{AM:MB} = 1:2. If A has coordinates (-1, -3) and B has coordinates (8, 9), the coordinates of M are

(2, 1)

\left(\frac{5}{3}, 0\right)

(5, 5)

\left(\frac{23}{3}, 8\right)

In the diagram below of triangle ABC, \overline{AC} is extended through point C to point D, and \overline{BE} is drawn to \overline{AC}.

Which equation is always true?

m \angle 1 = m \angle 3 + m \angle 2

m \angle 5 = m \angle 3 - m \angle 2

m \angle 6 = m \angle 3 - m \angle 2

m \angle 7 = m \angle 3 + m \angle 2

In the diagram below of right triangle ABC, AC = 8, and AB = 17.

Which equation would determine the value of angle A?

\sin A = \frac{8}{17}

\tan A = \frac{8}{15}

\cos A = \frac{15}{17}

\tan A = \frac{15}{8}

Francisco needs the three pieces of glass shown below to complete a stained glass window. The shapes, two triangles and a trapezoid, are measured in inches.

Glass can be purchased in rectangular sheets that are 12 inches wide. What is the minimum length of a sheet of glass, in inches, that Francisco must purchase in order to have enough to complete the window?

20

25

29

34

In the diagram of quadrilateral NAVY below, m \angle YNA = 30^\circ, m \angle YAN = 38^\circ, m \angle AVY = 94^\circ, and m \angle VAY = 46^\circ.

Which segment has the shortest length?

\overline{AY}

\overline{NY}

\overline{VA}

\overline{VY}

What is an equation of a circle whose center is (1,4) and diameter is 10?

x^{2} - 2x + y^{2} - 8y = 8

x^{2} + 2x + y^{2} + 8y = 8

x^{2} - 2x + y^{2} - 8y = 83

x^{2} + 2x + y^{2} + 8y = 83

On the set of axes below, \triangle ABC, altitude \overline{CG}, and median \overline{CM} are drawn.

Which expression represents the area of \triangle ABC?

\frac{(BC)(AC)}{2}

\frac{(GC)(BC)}{2}

\frac{(CM)(AB)}{2}

\frac{(GC)(AB)}{2}

In right triangle ABC, m\angle C = 90^\circ and AC \neq BC. Which trigonometric ratio is equivalent to \sin B?

cos A 

cos B 

tan A 

tan B 

As shown in the diagram below, the radius of a cone is 2.5 cm and its slant height is 6.5 cm.

How many cubic centimeters are in the volume of the cone?

12.5\pi

13.5\pi

30.0\pi

37.5\pi

What is an equation of the image of the line y = \frac{3}{2}x - 4 after a dilation of a scale factor of \frac{3}{4} centered at the origin?

y = \frac{9}{8}x - 4

y = \frac{9}{8}x - 3

y = \frac{3}{2}x - 4

y = \frac{3}{2}x - 3

PART II
Answer all 7 questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 

Write an equation of the line that is parallel to the line whose equation is 3y + 7 = 2x and passes through the point (2,6).

Parallelogram ABCD is adjacent to rhombus DEFG, as shown below, and \overline{FC} intersects \overline{AGD } at H.

If m\angle B = 118^\circ and m\angle AHC = 138^\circ, determine and state m\angle GFH.

As shown in the diagram below, secants \overrightarrow{PWR} and \overrightarrow{PTS} are drawn to circle O from external point P.

If m\angle RPS = 35^\circ and m\overset{\frown} {RS} = 121^\circ, determine and state m\overset{\frown}{WT}.

On the set of axes below, \triangle ABC is graphed with coordinates A(-2,-1), B(3,-1), and C(-2,-4). Triangle QRS, the image of \triangle ABC, is graphed with coordinates Q(-5,2), R(-5,7), and S(-8,2).

Describe a sequence of transformations that would map △ABC onto △QRS.

Given points A, B, and C, use a compass and straightedge to construct point D so that ABCD is a parallelogram.

[Leave all construction marks.]

On the set of axes below, \triangle DEF has vertices at the coordinates D(1,-1), E(3,4), and F(4,2), and point G has coordinates (3,1). Owen claims the median from point E must pass through point G.

Is Owen correct? Explain why.

A walking path at a local park is modeled on the grid below, where the length of each grid square is 10 feet. The town needs to submit paperwork to pave the walking path. Determine and state, to the nearest square foot, the area of the walking path.

PART III

Answer all 3 questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 

A triangle has vertices A(-2, 4), B(6, 2), and C(1, -1).

Prove that \triangle ABC is an isosceles right triangle.
[The use of the set of axes below is optional.]

Theresa has a rectangular pool 30 ft long, 15 ft wide, and 4 ft deep. Theresa fills her pool using city water at a rate of $3.95 per 100 gallons of water.

Nancy has a circular pool with a diameter of 24 ft and a depth of 4 ft. Nancy fills her pool with a water delivery service at a rate of $200 per 6000 gallons.

If Theresa and Nancy both fill their pools 6 inches from the top of the pool, determine and state who paid more to fill her pool. [1 ft3 water = 7.48 gallons]

As modeled in the diagram below, an access ramp starts on flat ground and ends at the beginning of the top step. Each step is 6 inches tall and 8 inches deep.

If the angle of elevation of the ramp is 4.76^\circ, determine and state the length of the ramp, to the nearest tenth of a foot. 

Length of the ramp: _______ feet

Determine and state, to the nearest tenth of a foot, the horizontal distance, d, from the bottom of the stairs to the bottom of the ramp.

Horizontal distance: _______ feet

PART IV

Answer the question in this part. A correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for the question to determine your answer. Note that diagrams are not necessarily drawn to scale. For the question in this part, a correct numerical answer with no work shown will receive only 1 credit. 

In the diagram of quadrilateral ABCD with diagonal \overline{AC} shown below, segments GH and EF are drawn, \overline{AE} \cong \overline{CG}, \overline{BE}\cong\overline{DG}, \overline{AH} \cong\overline{CF}, and \overline{AD}\cong\overline{CB}.

Prove: \overline{EF} \cong\overline{GH}

After a dilation with center (0,0), the image of \overline{DB} is \overline{D'B'}. If DB = 4.5 and D'B' = 18, the scale factor of this dilation is:

\frac{1}{5}

5

\frac{1}{4}

4

In the diagram below, \triangle ABC with sides of 13, 15, and 16, is mapped onto \triangle DEF after a clockwise rotation of 90° about point P.

If DE = 2x - 1, what is the value of x?

7

7.5

8

8.5

On the set of axes below, \triangle ABC has vertices at A(-2,0), B(2,-4), C(4,2), and \triangle DEF has vertices at D(4,0), E(-4,8), F(-8,-4).

Which sequence of transformations will map △ABC onto △DEF?

a dilation of △ABC by a scale factor of 2 centered at point A

a dilation of △ABC by a scale factor of \frac{1}{2} centered at point A

a dilation of △ABC by a scale factor of 2 centered at the origin, followed by a rotation of 180^\circ about the origin

a dilation of △ABC by a scale factor of \frac{1}{2} centered at the origin, followed by a rotation of 180^\circ about the origin

The figure below shows a rhombus with noncongruent diagonals. 

Which transformation would not carry this rhombus onto itself?

a reflection over the shorter diagonal

a reflection over the longer diagonal

a clockwise rotation of 90^\circ about the intersection of the diagonals

a counterclockwise rotation of 180^\circ about the intersection of the diagonals

In the diagram below of circle O, points K, A, T, I, and E are on the circle,  \triangle KAE  and  \triangle ITE  are drawn,  \overset{\frown}{KE}\cong\overset{\frown}{EI}, and  \angle EKA \cong\angle EIT. 

Which statement about \triangle KAE and \triangle ITE is always true?

They are neither congruent nor similar.

They are similar but not congruent.

They are right triangles.

They are congruent.

In right triangle ABC shown below, point D is on \overline{AB} and point E is on \overline{CB} such that \overline{AC} \parallel \overline{DE}.

If AB = 15, BC = 12, and EC = 7, what is the length of \overline{BD}?

8.75

6.25

5

4

In rhombus VENU, diagonals \overline{VN} and \overline{EU} intersect at S. If VN = 12 and EU = 16, what is the perimeter of the rhombus?

80

40

20

10

Given right triangle ABC with a right angle at C, m\angle B = 61^\circ.
Given right triangle RST with a right angle at T, m\angle R = 29^\circ.


Which proportion in relation to \triangle ABC and \triangle RST is not correct?

\frac{AB}{RS} = \frac{RT}{AC}

\frac{BC}{ST} = \frac{AB}{RS}

\frac{BC}{ST} = \frac{AC}{RT}

\frac{AB}{AC} = \frac{RS}{RT}

A vendor is using an 8-ft by 8-ft tent for a craft fair. The legs of the tent are 9 ft tall and the top forms a square pyramid with a height of 3 ft.

What is the volume, in cubic feet, of space the tent occupies?

256

640

672

768

In the diagram below of right triangle KMI, altitude \overline{IG} is drawn to hypotenuse \overline{KM}. 

If KG = 9 and IG = 12, the length of \overline{IM} is

15

16

20

25

Which three-dimensional figure will result when a rectangle 6 inches long and 5 inches wide is continuously rotated about the longer side?

a rectangular prism with a length of 6 inches, width of 6 inches, and height of 5 inches

a rectangular prism with a length of 6 inches, width of 5 inches, and height of 5 inches

a cylinder with a radius of 5 inches and a height of 6 inches

a cylinder with a radius of 6 inches and a height of 5 inches

Which statement about parallelograms is always true?

The diagonals are congruent.

The diagonals bisect each other.

The diagonals are perpendicular.

The diagonals bisect their respective angles.

From a point on the ground one-half mile from the base of a historic monument, the angle of elevation to its top is 11.87°. To the nearest foot, what is the height of the monument?

543

555

1086

1110

The area of a sector of a circle with a radius measuring 15 cm is 75\pi \, \text{cm}^2. What is the measure of the central angle that forms the sector?

72°

120°

144°

180°

Point M divides \overline{AB} so that \text{AM:MB} = 1:2. If A has coordinates (-1, -3) and B has coordinates (8, 9), the coordinates of M are

(2, 1)

\left(\frac{5}{3}, 0\right)

(5, 5)

\left(\frac{23}{3}, 8\right)

In the diagram below of triangle ABC, \overline{AC} is extended through point C to point D, and \overline{BE} is drawn to \overline{AC}.

Which equation is always true?

m \angle 1 = m \angle 3 + m \angle 2

m \angle 5 = m \angle 3 - m \angle 2

m \angle 6 = m \angle 3 - m \angle 2

m \angle 7 = m \angle 3 + m \angle 2

In the diagram below of right triangle ABC, AC = 8, and AB = 17.

Which equation would determine the value of angle A?

\sin A = \frac{8}{17}

\tan A = \frac{8}{15}

\cos A = \frac{15}{17}

\tan A = \frac{15}{8}

Francisco needs the three pieces of glass shown below to complete a stained glass window. The shapes, two triangles and a trapezoid, are measured in inches.

Glass can be purchased in rectangular sheets that are 12 inches wide. What is the minimum length of a sheet of glass, in inches, that Francisco must purchase in order to have enough to complete the window?

20

25

29

34

In the diagram of quadrilateral NAVY below, m \angle YNA = 30^\circ, m \angle YAN = 38^\circ, m \angle AVY = 94^\circ, and m \angle VAY = 46^\circ.

Which segment has the shortest length?

\overline{AY}

\overline{NY}

\overline{VA}

\overline{VY}

What is an equation of a circle whose center is (1,4) and diameter is 10?

x^{2} - 2x + y^{2} - 8y = 8

x^{2} + 2x + y^{2} + 8y = 8

x^{2} - 2x + y^{2} - 8y = 83

x^{2} + 2x + y^{2} + 8y = 83

On the set of axes below, \triangle ABC, altitude \overline{CG}, and median \overline{CM} are drawn.

Which expression represents the area of \triangle ABC?

\frac{(BC)(AC)}{2}

\frac{(GC)(BC)}{2}

\frac{(CM)(AB)}{2}

\frac{(GC)(AB)}{2}

In right triangle ABC, m\angle C = 90^\circ and AC \neq BC. Which trigonometric ratio is equivalent to \sin B?

cos A 

cos B 

tan A 

tan B 

As shown in the diagram below, the radius of a cone is 2.5 cm and its slant height is 6.5 cm.

How many cubic centimeters are in the volume of the cone?

12.5\pi

13.5\pi

30.0\pi

37.5\pi

What is an equation of the image of the line y = \frac{3}{2}x - 4 after a dilation of a scale factor of \frac{3}{4} centered at the origin?

y = \frac{9}{8}x - 4

y = \frac{9}{8}x - 3

y = \frac{3}{2}x - 4

y = \frac{3}{2}x - 3

NYSED High School Regents Geometry

NYSED High School Regents Geometry

Part I

PART II

Write an equation of the line that is parallel to the line whose equation is 3y + 7 = 2x and passes through the point (2,6).

Parallelogram ABCD is adjacent to rhombus DEFG, as shown below, and \overline{FC} intersects \overline{AGD } at H.If m\angle B = 118^\circ and m\angle AHC = 138^\circ, determine and state m\angle GFH.

As shown in the diagram below, secants \overrightarrow{PWR} and \overrightarrow{PTS} are drawn to circle O from external point P.If m\angle RPS = 35^\circ and m\overset{\frown} {RS} = 121^\circ, determine and state m\overset{\frown}{WT}.

On the set of axes below, \triangle ABC is graphed with coordinates A(-2,-1), B(3,-1), and C(-2,-4). Triangle QRS, the image of \triangle ABC, is graphed with coordinates Q(-5,2), R(-5,7), and S(-8,2).Describe a sequence of transformations that would map △ABC onto △QRS.

Given points A, B, and C, use a compass and straightedge to construct point D so that ABCD is a parallelogram.[Leave all construction marks.]

On the set of axes below, \triangle DEF has vertices at the coordinates D(1,-1), E(3,4), and F(4,2), and point G has coordinates (3,1). Owen claims the median from point E must pass through point G.Is Owen correct? Explain why.

A walking path at a local park is modeled on the grid below, where the length of each grid square is 10 feet. The town needs to submit paperwork to pave the walking path. Determine and state, to the nearest square foot, the area of the walking path.

PART III

A triangle has vertices A(-2, 4), B(6, 2), and C(1, -1).Prove that \triangle ABC is an isosceles right triangle. [The use of the set of axes below is optional.]

PART IV

In the diagram of quadrilateral ABCD with diagonal \overline{AC} shown below, segments GH and EF are drawn, \overline{AE} \cong \overline{CG}, \overline{BE}\cong\overline{DG}, \overline{AH} \cong\overline{CF}, and \overline{AD}\cong\overline{CB}.Prove: \overline{EF} \cong\overline{GH}

Part I

After a dilation with center (0,0), the image of \overline{DB} is \overline{D'B'}. If DB = 4.5 and D'B' = 18, the scale factor of this dilation is:

In the diagram below, \triangle ABC with sides of 13, 15, and 16, is mapped onto \triangle DEF after a clockwise rotation of 90° about point P.If DE = 2x - 1, what is the value of x?

On the set of axes below, \triangle ABC has vertices at A(-2,0), B(2,-4), C(4,2), and \triangle DEF has vertices at D(4,0), E(-4,8), F(-8,-4).Which sequence of transformations will map △ABC onto △DEF?

The figure below shows a rhombus with noncongruent diagonals. Which transformation would not carry this rhombus onto itself?

In the diagram below of circle O, points K, A, T, I, and E are on the circle, \triangle KAE and \triangle ITE are drawn, \overset{\frown}{KE}\cong\overset{\frown}{EI}, and \angle EKA \cong\angle EIT. Which statement about \triangle KAE and \triangle ITE is always true?

In right triangle ABC shown below, point D is on \overline{AB} and point E is on \overline{CB} such that \overline{AC} \parallel \overline{DE}.If AB = 15, BC = 12, and EC = 7, what is the length of \overline{BD}?

In rhombus VENU, diagonals \overline{VN} and \overline{EU} intersect at S. If VN = 12 and EU = 16, what is the perimeter of the rhombus?

Given right triangle ABC with a right angle at C, m\angle B = 61^\circ. Given right triangle RST with a right angle at T, m\angle R = 29^\circ.Which proportion in relation to \triangle ABC and \triangle RST is not correct?

A vendor is using an 8-ft by 8-ft tent for a craft fair. The legs of the tent are 9 ft tall and the top forms a square pyramid with a height of 3 ft.What is the volume, in cubic feet, of space the tent occupies?

In the diagram below of right triangle KMI, altitude \overline{IG} is drawn to hypotenuse \overline{KM}. If KG = 9 and IG = 12, the length of \overline{IM} is

Which three-dimensional figure will result when a rectangle 6 inches long and 5 inches wide is continuously rotated about the longer side?

Which statement about parallelograms is always true?

From a point on the ground one-half mile from the base of a historic monument, the angle of elevation to its top is 11.87°. To the nearest foot, what is the height of the monument?

The area of a sector of a circle with a radius measuring 15 cm is 75\pi \, \text{cm}^2. What is the measure of the central angle that forms the sector?

Point M divides \overline{AB} so that \text{AM:MB} = 1:2. If A has coordinates (-1, -3) and B has coordinates (8, 9), the coordinates of M are

In the diagram below of triangle ABC, \overline{AC} is extended through point C to point D, and \overline{BE} is drawn to \overline{AC}.Which equation is always true?

In the diagram below of right triangle ABC, AC = 8, and AB = 17.Which equation would determine the value of angle A?

In the diagram of quadrilateral NAVY below, m \angle YNA = 30^\circ, m \angle YAN = 38^\circ, m \angle AVY = 94^\circ, and m \angle VAY = 46^\circ.Which segment has the shortest length?

What is an equation of a circle whose center is (1,4) and diameter is 10?

On the set of axes below, \triangle ABC, altitude \overline{CG}, and median \overline{CM} are drawn.Which expression represents the area of \triangle ABC?

In right triangle ABC, m\angle C = 90^\circ and AC \neq BC. Which trigonometric ratio is equivalent to \sin B?

As shown in the diagram below, the radius of a cone is 2.5 cm and its slant height is 6.5 cm.How many cubic centimeters are in the volume of the cone?

What is an equation of the image of the line y = \frac{3}{2}x - 4 after a dilation of a scale factor of \frac{3}{4} centered at the origin?

PART II

Write an equation of the line that is parallel to the line whose equation is 3y + 7 = 2x and passes through the point (2,6).

Parallelogram ABCD is adjacent to rhombus DEFG, as shown below, and \overline{FC} intersects \overline{AGD } at H.If m\angle B = 118^\circ and m\angle AHC = 138^\circ, determine and state m\angle GFH.

As shown in the diagram below, secants \overrightarrow{PWR} and \overrightarrow{PTS} are drawn to circle O from external point P.If m\angle RPS = 35^\circ and m\overset{\frown} {RS} = 121^\circ, determine and state m\overset{\frown}{WT}.

On the set of axes below, \triangle ABC is graphed with coordinates A(-2,-1), B(3,-1), and C(-2,-4). Triangle QRS, the image of \triangle ABC, is graphed with coordinates Q(-5,2), R(-5,7), and S(-8,2).Describe a sequence of transformations that would map △ABC onto △QRS.

Given points A, B, and C, use a compass and straightedge to construct point D so that ABCD is a parallelogram.[Leave all construction marks.]

On the set of axes below, \triangle DEF has vertices at the coordinates D(1,-1), E(3,4), and F(4,2), and point G has coordinates (3,1). Owen claims the median from point E must pass through point G.Is Owen correct? Explain why.

A walking path at a local park is modeled on the grid below, where the length of each grid square is 10 feet. The town needs to submit paperwork to pave the walking path. Determine and state, to the nearest square foot, the area of the walking path.

PART III

A triangle has vertices A(-2, 4), B(6, 2), and C(1, -1).Prove that \triangle ABC is an isosceles right triangle. [The use of the set of axes below is optional.]

PART IV

In the diagram of quadrilateral ABCD with diagonal \overline{AC} shown below, segments GH and EF are drawn, \overline{AE} \cong \overline{CG}, \overline{BE}\cong\overline{DG}, \overline{AH} \cong\overline{CF}, and \overline{AD}\cong\overline{CB}.Prove: \overline{EF} \cong\overline{GH}

Parallelogram ABCD is adjacent to rhombus DEFG, as shown below, and \overline{FC} intersects \overline{AGD } at H.

If m\angle B = 118^\circ and m\angle AHC = 138^\circ, determine and state m\angle GFH.

As shown in the diagram below, secants \overrightarrow{PWR} and \overrightarrow{PTS} are drawn to circle O from external point P.

If m\angle RPS = 35^\circ and m\overset{\frown} {RS} = 121^\circ, determine and state m\overset{\frown}{WT}.

On the set of axes below, \triangle ABC is graphed with coordinates A(-2,-1), B(3,-1), and C(-2,-4). Triangle QRS, the image of \triangle ABC, is graphed with coordinates Q(-5,2), R(-5,7), and S(-8,2).

Describe a sequence of transformations that would map △ABC onto △QRS.

Given points A, B, and C, use a compass and straightedge to construct point D so that ABCD is a parallelogram.

[Leave all construction marks.]

On the set of axes below, \triangle DEF has vertices at the coordinates D(1,-1), E(3,4), and F(4,2), and point G has coordinates (3,1). Owen claims the median from point E must pass through point G.

Is Owen correct? Explain why.

A triangle has vertices A(-2, 4), B(6, 2), and C(1, -1).

Prove that \triangle ABC is an isosceles right triangle.
[The use of the set of axes below is optional.]

In the diagram of quadrilateral ABCD with diagonal \overline{AC} shown below, segments GH and EF are drawn, \overline{AE} \cong \overline{CG}, \overline{BE}\cong\overline{DG}, \overline{AH} \cong\overline{CF}, and \overline{AD}\cong\overline{CB}.

Prove: \overline{EF} \cong\overline{GH}

In the diagram below, \triangle ABC with sides of 13, 15, and 16, is mapped onto \triangle DEF after a clockwise rotation of 90° about point P.

If DE = 2x - 1, what is the value of x?

On the set of axes below, \triangle ABC has vertices at A(-2,0), B(2,-4), C(4,2), and \triangle DEF has vertices at D(4,0), E(-4,8), F(-8,-4).

Which sequence of transformations will map △ABC onto △DEF?

The figure below shows a rhombus with noncongruent diagonals.

Which transformation would not carry this rhombus onto itself?

In the diagram below of circle O, points K, A, T, I, and E are on the circle, \triangle KAE and \triangle ITE are drawn, \overset{\frown}{KE}\cong\overset{\frown}{EI}, and \angle EKA \cong\angle EIT.

Which statement about \triangle KAE and \triangle ITE is always true?

In right triangle ABC shown below, point D is on \overline{AB} and point E is on \overline{CB} such that \overline{AC} \parallel \overline{DE}.

If AB = 15, BC = 12, and EC = 7, what is the length of \overline{BD}?

Given right triangle ABC with a right angle at C, m\angle B = 61^\circ. Given right triangle RST with a right angle at T, m\angle R = 29^\circ.

Which proportion in relation to \triangle ABC and \triangle RST is not correct?

A vendor is using an 8-ft by 8-ft tent for a craft fair. The legs of the tent are 9 ft tall and the top forms a square pyramid with a height of 3 ft.

What is the volume, in cubic feet, of space the tent occupies?

In the diagram below of right triangle KMI, altitude \overline{IG} is drawn to hypotenuse \overline{KM}.

If KG = 9 and IG = 12, the length of \overline{IM} is

In the diagram below of triangle ABC, \overline{AC} is extended through point C to point D, and \overline{BE} is drawn to \overline{AC}.

Which equation is always true?

In the diagram below of right triangle ABC, AC = 8, and AB = 17.

Which equation would determine the value of angle A?

In the diagram of quadrilateral NAVY below, m \angle YNA = 30^\circ, m \angle YAN = 38^\circ, m \angle AVY = 94^\circ, and m \angle VAY = 46^\circ.

Which segment has the shortest length?

On the set of axes below, \triangle ABC, altitude \overline{CG}, and median \overline{CM} are drawn.

Which expression represents the area of \triangle ABC?

As shown in the diagram below, the radius of a cone is 2.5 cm and its slant height is 6.5 cm.

How many cubic centimeters are in the volume of the cone?

Parallelogram ABCD is adjacent to rhombus DEFG, as shown below, and \overline{FC} intersects \overline{AGD } at H.

If m\angle B = 118^\circ and m\angle AHC = 138^\circ, determine and state m\angle GFH.

As shown in the diagram below, secants \overrightarrow{PWR} and \overrightarrow{PTS} are drawn to circle O from external point P.

If m\angle RPS = 35^\circ and m\overset{\frown} {RS} = 121^\circ, determine and state m\overset{\frown}{WT}.

On the set of axes below, \triangle ABC is graphed with coordinates A(-2,-1), B(3,-1), and C(-2,-4). Triangle QRS, the image of \triangle ABC, is graphed with coordinates Q(-5,2), R(-5,7), and S(-8,2).

Describe a sequence of transformations that would map △ABC onto △QRS.

Given points A, B, and C, use a compass and straightedge to construct point D so that ABCD is a parallelogram.

[Leave all construction marks.]

On the set of axes below, \triangle DEF has vertices at the coordinates D(1,-1), E(3,4), and F(4,2), and point G has coordinates (3,1). Owen claims the median from point E must pass through point G.

Is Owen correct? Explain why.

A triangle has vertices A(-2, 4), B(6, 2), and C(1, -1).

Prove that \triangle ABC is an isosceles right triangle.
[The use of the set of axes below is optional.]

In the diagram of quadrilateral ABCD with diagonal \overline{AC} shown below, segments GH and EF are drawn, \overline{AE} \cong \overline{CG}, \overline{BE}\cong\overline{DG}, \overline{AH} \cong\overline{CF}, and \overline{AD}\cong\overline{CB}.

Prove: \overline{EF} \cong\overline{GH}