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This page shows questions in the Geometry Reasoning public release module at MSDE. Geometry
"Geometry Reasoning"

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Expand All Official Practice Test

This is a question with 2 parts, including a test question that allows you to enter extended text in your response.

The coordinates of the vertices of quadrilateral PQRS are shown in the xy-plane.

The figure shows quadrilateral P Q R S, in the x y plane, with the origin labeled, O. The numbers 1, through 10, are indicated on both axes. The coordinates of the vertices are as follows,.Point P: 4 comma 10,.  Point Q: 10 comma 7,.  Point R: 6 comma 4,.  Point S: 4 comma 5.

Part A

Prove that quadrilateral PQRS is a trapezoid.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Part B

Determine whether quadrilateral PQRS is an isosceles trapezoid. Show your work or explain your answer.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

There is no alt-text. (38 files were searched.)

Holistic Rubric for 4-Point Reasoning Constructed Response Items

Sample Top Score Response

Part A

Calculating the slopes:

The slope of side P Q : 10 − 7 4 − 10 = 3 − 6 = − 1 2

The slope of side R S : 5 − 4 4 − 6 = 1 − 2

The slope of side P S : 10 − 5 4 − 4 = 5 0 which is undefined

The slope of side Q R : 7 − 4 10 − 6 = 3 4

P Q R S is a trapezoid because sides P Q and R S are parallel and sides P S and Q R are not parallel.

Part B

Calculating the side lengths:

P S = ( 10 − 5 ) 2 + ( 4 − 4 ) 2 = 5

Q R = ( 7 − 4 ) 2 + ( 10 − 6 ) 2 = 5

Since the side lengths are equal, P Q R S is an isosceles trapezoid.

Scoring

Points

Sample Characteristics

4 Points

A four-point response for reasoning items provides evidence of correct, complete, and appropriate mathematical reasoning.

The response may:

  • be clear and well developed with logical reasoning communicated by the use of precise and appropriate representations, symbols, drawings, or mathematical vocabulary.
  • demonstrate a thorough understanding of the mathematics.
  • contain minor flaws that do not detract from the correct reasoning or demonstration of a thorough understanding.

3 Points

A three-point response for reasoning items provides evidence of essentially correct, essentially complete, and essentially appropriate mathematical reasoning.

The response may:

  • be clear and developed with logical reasoning communicated by the use of essentially precise and appropriate representations, symbols, drawings, or mathematical vocabulary.
  • contain minor flaws that detract from the correct reasoning or demonstration of a thorough understanding.

2 Points

A two-point response for reasoning items provides evidence of partially correct mathematical reasoning.

The response may:

  • display an incomplete reasoning process.
  • contain mathematical flaws.

1 Point

A one-point response for reasoning items provides limited evidence of correct mathematical reasoning.

The response may:

  • demonstrate the beginning of a valid chain of reasoning.
  • reflect a lack of essential understanding of the underlying mathematical concepts.
  • contain the correct solution, but work is limited or missing.
  • contain errors in the fundamental mathematical procedures or reasoning.
  • contain omissions or irregularities that lead to an inadequate solution.

0 Point

A zero-point response is completely incorrect, incoherent or irrelevant.

This is a multiple choice question that allows you to select only one option.

A cone has a base radius of 3 centimeters and a height of 5 centimeters. A student correctly calculates its volume to be 15π cubic centimeters.

The figure shows a cone. The radius of the base is labeled 3 centimeters, and the height is labeled 5 centimeters.

The student thinks that a simpler formula for the volume of a cone is V=πrh because π(3)(5)=15π.

Which statement explains the conditions for which the student’s claim would be true?

  1. A.
    The claim is true only when the height is 5.
  2. B.
    The claim is true only when the radius of the cone is 3.
  3. C.
    The claim is true regardless of the dimensions of the cone .
  4. D.
    The claim is true whenever the product of the base and height is 15.
i2

Correcti1

The formula does not always work.i3

The revised formula is not affected by the height.i4

The revised formula would still work if the height was 6 and therefore the product would be 18.

This is a test question that allows you to enter extended text in your response.

Right triangle XYZ is shown in the xy-plane. Vertex X has coordinates (2,3). The length of segment X Y is 2 units, and the length of segment X Z is 6 units.

The figure shows right triangle X Y Z, in the x y plane, with the origin labeled, O,. Vertex X has coordinates, 2, comma, 3. Vertex Y is above vertex X, and vertex Z is to the right of vertex X, such that angle Y X Z, is a right angle. The length of segment X Y, is 2 units, and the length of segment X Z, is 6 units.

A student’s work for finding the slope of the perpendicular bisector of segment Y Z is shown.

The slope of segment Y Z is 2 sixths, or 1 third. The opposite of the reciprocal of 1 third is negative 3. So, the perpendicular bisector of segment Y Z has a slope of negative 3.
  • Describe the student’s mistake.
  • Find the equation of the line that represents the perpendicular bisector of segment Y Z. Show your work or explain how you found the equation.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

There is no alt-text. (38 files were searched.)

Holistic Rubric for 4-Point Reasoning Constructed Response Items

Sample Top Score Response

The student’s mistake was using a slope of 1 3 instead of - 1 3 since side Y Z points down and to the right.

The opposite of the reciprocal of - 1 3 is 3 .

The perpendicular bisector of side Y Z passes through the midpoint of side Y Z . The coordinates of the midpoint are x = 2 + 6 2 = 5 y = 3 + 2 2 = 4 .

The equation of the perpendicular bisector is x − 4 = 3 ( x − 5 ) y = 3 x − 15 + 4 y = 3 x − 11 .

Scoring

Points

Sample Characteristics

4 Points

A four-point response for reasoning items provides evidence of correct, complete, and appropriate mathematical reasoning.

The response may:

  • be clear and well developed with logical reasoning communicated by the use of precise and appropriate representations, symbols, drawings, or mathematical vocabulary.
  • demonstrate a thorough understanding of the mathematics.
  • contain minor flaws that do not detract from the correct reasoning or demonstration of a thorough understanding.

3 Points

A three-point response for reasoning items provides evidence of essentially correct, essentially complete, and essentially appropriate mathematical reasoning.

The response may:

  • be clear and developed with logical reasoning communicated by the use of essentially precise and appropriate representations, symbols, drawings, or mathematical vocabulary.
  • contain minor flaws that detract from the correct reasoning or demonstration of a thorough understanding.

2 Points

A two-point response for reasoning items provides evidence of partially correct mathematical reasoning.

The response may:

  • display an incomplete reasoning process.
  • contain mathematical flaws.

1 Point

A one-point response for reasoning items provides limited evidence of correct mathematical reasoning.

The response may:

  • demonstrate the beginning of a valid chain of reasoning.
  • reflect a lack of essential understanding of the underlying mathematical concepts.
  • contain the correct solution, but work is limited or missing.
  • contain errors in the fundamental mathematical procedures or reasoning.
  • contain omissions or irregularities that lead to an inadequate solution.

0 Point

A zero-point response is completely incorrect, incoherent or irrelevant.

This is a multiple choice question that allows you to select several options.

A student claims that any quadrilateral with two right angles must be a rectangle.

Which figures can be used to show the student is incorrect?

Select all that apply.

  1. A.
    The figure shows a quadrilateral with top and bottom sides that appear horizontal and left and right sides that appear vertical. Right angle symbols are shown in the lower left and upper right corners. A single hash mark is shown on each of the four sides.
  2. B.
    The figure shows a quadrilateral with top and bottom sides that appear horizontal, a left side that appears vertical, and a right side that is slanted. Right angle symbols are shown in the lower left and upper left corners. A single hash mark is shown on the top and right sides.
  3. C.
    The figure shows a quadrilateral with top and bottom sides that appear horizontal and left and right sides that appear vertical. Right angle symbols are shown in the lower left and upper left corners. A single hash mark is shown on the left and right sides. Double hash marks are shown on the top and bottom sides.
  4. D.
    The figure shows a quadrilateral with no horizontal or vertical sides. The upper left and lower right sides slant upward and to the right, while the upper right and lower left sides slant upward and to the left. Right angle symbols are shown in the left and right corners. A single hash mark is shown on the upper left and upper right sides. Double hash marks are shown on the lower left and lower right sides.
  5. E.
    The figure shows a quadrilateral with a bottom side that appears horizontal, left and right sides that appear vertical, and a top side that is slanted. Right angle symbols are shown in the lower left and lower right corners. A single hash mark is shown on the bottom and right sides.

This is a multiple choice question that allows you to select several options.

Five points are shown on a number line.

The figure shows five points, plotted at uneven intervals on a number line. No numbering is indicated on the number line. The points are labeled P, Q, R, S, and T, from left to right.

Information about some of the lengths of different segments is given.

  • The length of line segment P Q is 5 units.
  • The length of line segment R S is equal to the length of line segment P R.
  • The length of line segment S T is 3 times the length of line segment Q R.
  • The length of line segment P T is 25 units.

Which statements are correct?

Select all that apply.

  1. A.
    The length of line segment Q R is 3 units.
  2. B.
    The length of line segment R S is 8 units.
  3. C.
    The length of line segment Q S is 19 units.
  4. D.
    The length of line segment R T is 13 units.
  5. E.
    The length of line segment S T is 9 units.
i1

Correct i2

Correct i5

Correct i3

Incorrectly calculated Q R to be 7 i4

Incorrectly calculated Q R to be 2

This is a question with 2 parts, including a question with drop-down menus from which you must select an option to fill in the blank.

In triangle K L M and triangle X Y Z shown, segment K L is congruent to segment X Y and angle L is congruent to angle Y.

The figure shows triangle K L M, and triangle X Y Z,.  A single hash mark is shown on segment K L, and segment X Y,.  A single arc is shown at angle L, and angle Y.

What additional information is needed to prove that triangle K L M is congruent to triangle X Y Z?

Select from the drop-down menus to correctly complete each statement.

If segment L M is congruent to segment Y Z, then triangle K L M is congruent to triangle X Y Z by the  
  postulate.
If angle M is congruent to angle Z, then triangle K L M is congruent to triangle X Y Z by the  
  postulate.