Section B Practice Problems

Problem 1

Determine whether the statement is true or false. Explain or show your reasoning.

  1. The shape is a rectangle.
  2. The shape is a square.
  3. The shape is a rhombus.

Quadrilateral on grid. all sides equal length. no right angles. 

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Solution
  1. False. Sample response: The angles are not right angles so it is not a rectangle.
  2. False. Sample response: The sides are the same length but the angles are not right angles so it is not a square.
  3. True. Sample response: It is a rhombus because all 4 sides are the same length.

Problem 2

  1. Draw a trapezoid that is also a parallelogram. Explain how you know it is a trapezoid and a parallelogram.

    blank grid

  2. Draw a trapezoid that is not a parallelogram. Explain how you know it is a trapezoid but not a parallelogram.

    blank grid

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Solution
Sample response:

  1. quadrilateral on grid
    It is a trapezoid because the two horizontal sides are parallel. It is a parallelogram because the lines containing the other two sides are also parallel.

  2. quadrilateral on grid
    It is a trapezoid because the two horizontal sides are parallel. It is not a parallelogram because the vertical side is not parallel to the slanted side.

Problem 3

Determine if it’s possible to draw each shape so it contains the 2 sides drawn on the grid. Explain your reasoning.

  1. a square
  2. a rectangle
  3. a rhombus

2 line segments of equal length on grid. do not meet at a right angle.

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Solution
  1. No. Sample response: The angle made by sides that are given is not a right angle.
  2. No. Sample response: The same reason as for the square.
  3. Yes. Sample response: I drew a quadrilateral with 4 equal sides so it's a rhombus.
    quadrilateral on grid

Problem 4

Decide if each statement is true or false. Explain or show your reasoning.

  1. A parallelogram is sometimes a rhombus.
  2. A rhombus is always a parallelogram.
  3. A trapezoid is never a rectangle.
  4. A rectangle is never a square.
  5. A parallelogram is always a trapezoid.

Show Solution
Solution
  1. True. Sample response: When the four sides of a parallelogram are equal, then it is a rhombus.
  2. True. Sample response: The opposite sides of a rhombus are always contained in parallel lines.
  3. False. Sample response: A trapezoid can be a rectangle if it has 4 right angles.
  4. False. Sample response: A rectangle is a square when it has 4 equal sides.
  5. True. Sample response: A parallelogram always has 2 pairs of parallel sides and a trapezoid has at least one pair of parallel sides. (Note: this is false if the definition of trapezoid is a quadrilateral with exactly one pair of parallel sides)

Problem 5

Draw a right triangle on the grid to match each description. If you think there is no such triangle, explain why.

  1. It has 2 equal side lengths.

    blank grid

  2. It has 3 equal side lengths.

    blank grid

  3. It has 3 different side lengths.

    blank grid

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Solution
Sample responses:
  1. See sample drawing.
  2. This is not possible. The angles in a triangle with 3 equal sides are never right angles.
  3. See sample drawing.
triangles on grid

Problem 6

  1. Jada cut a quadrilateral in half, from 1 vertex to the opposite vertex, and she got 2 triangles that each have exactly 2 equal side lengths. What kind of quadrilateral could Jada have cut in half? Explain or show your reasoning.

  2. Elena put together 2 right triangles to make a quadrilateral. What kind of quadrilateral could Elena have made? Explain or show your reasoning.
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Solution
Sample response:
  1. It could be a square or a rhombus. It could also have been a shape that does not fit any of the categories we looked at.
    quadrilateral on square grid
  2. She could get a rectangle or a square. She could also get a parallelogram or a trapezoid or a shape that does not fit any of the categories we studied.
    shapes on grid

Problem 7

  1. Can you draw a square on the grid that does not have a vertical or horizontal side? Explain or show your reasoning.

    blank grid

  2. Draw the line segment from (4,4)(4, 4) to (6,5)(6, 5). Can you draw a square that contains this segment as one of its sides? Explain or show your reasoning.

    Coordinate grid. horizontal and vertical axis, 0 to 10, by 1's.

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Solution
Sample responses:

  1. quadrilateral on grid

    This is a square because the four sides are equal and the four angles are right angles. The sides are equal because they all go from one corner of a grid square to the opposite corner. I sketched the shape on patty paper to check that the angles are all right angles.


  2. quadrilateral on coordinate plane

    Yes, this is a square because each side is the diagonal of a 2 unit by 1 unit rectangle. Then I checked that the angles are all 90 degrees.