2D Figure Classification

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Answer Key & Explanations

2D Figure Classification · Grade 5 · Worksheet 3

1. Sophia is creating a classification chart for her geometry project. She has four shapes: a square with sides of 508 mm, a rhombus with sides of 508 mm but no right angles, a rectangle that is 512 mm long and 508 mm wide, and a parallelogram with sides of 512 mm and 508 mm and no right angles. Her classmate Noah says, 'Since the square has all sides equal, it is a rhombus, but it is also a rectangle because it has four right angles. Therefore, a square is both a rhombus and a rectangle.' Is Noah correct? Explain your reasoning using the hierarchical relationships between these shapes. Answer: Yes, Noah is correct. Solution: Identify the properties of a square. A square has four equal sides and four right angles. Since the square has all sides equal (508 mm), it satisfies the condition for a rhombus.
   Full step-by-step solution

   Step 1: Identify the properties of a square. A square has four equal sides and four right angles. Step 2: Check if a square is a rhombus. A rhombus is defined as a quadrilateral with all four sides equal. Since the square has all sides equal (508 mm), it satisfies the condition for a rhombus. Therefore, every square is a rhombus. Step 3: Check if a square is a rectangle. A rectangle is defined as a quadrilateral with four right angles. Since the square has four right angles, it satisfies the condition for a rectangle. Therefore, every square is a rectangle. Step 4: Conclusion: Noah is correct. A square is both a rhombus and a rectangle because it possesses the defining properties of both shapes. This is a key part of the hierarchical classification: all squares are rhombuses, and all squares are rectangles, but not all rhombuses or rectangles are squares. The answer is: Yes, Noah is correct.

2. Emma has drawn a large quadrilateral with four sides. She notices that all four sides are the same length, but the shape is not a square because its angles are not 90 degrees. Later, she draws another quadrilateral where both pairs of opposite sides are parallel, and all angles are 90 degrees, but the sides are not all equal. What is the most specific name for Emma's first shape, and what is the most specific name for her second shape? Then, explain whether every shape like the first one is also a special type of the second shape. Answer: First shape: rhombus; second shape: rectangle; No, not every rhombus is a rectangle because a rhombus does not have to have right angles. Solution: Identify the first shape. It has four equal sides (all sides the same length) but no right angles. It has opposite sides parallel and all angles 90 degrees, but not all sides equal.
   Full step-by-step solution

   Step 1: Identify the first shape. It has four equal sides (all sides the same length) but no right angles. The most specific quadrilateral with all sides equal is a rhombus. (A square also has all sides equal, but it requires right angles, which this shape lacks.) So the first shape is a rhombus. Step 2: Identify the second shape. It has opposite sides parallel and all angles 90 degrees, but not all sides equal. A quadrilateral with opposite sides parallel is a parallelogram. Adding the condition of all right angles makes it a rectangle. Since the sides are not all equal, it cannot be a square. So the second shape is a rectangle. Step 3: Answer the hierarchy question. A rhombus has all sides equal, but its angles are not necessarily 90 degrees. A rectangle has all angles 90 degrees, but its sides are not necessarily equal. For a rhombus to be a rectangle, it would need to have all right angles, which is not guaranteed. Therefore, not every rhombus is a rectangle. (However, every square is both a rhombus and a rectangle.) The answer is: First shape = rhombus, second shape = rectangle. No, not every rhombus is a rectangle.

3. Isabella is helping her art teacher organize a display of geometric shapes. She has a bin labeled 'Quadrilaterals' that contains a rhombus, a square, a trapezoid, and a rectangle. The teacher asks her to move any shape that is also a parallelogram into a new bin. Which shapes from the 'Quadrilaterals' bin should Isabella move into the 'Parallelogram' bin? Explain your reasoning. Answer: Square, rectangle, and rhombus Solution: Recall the definition of a parallelogram: a quadrilateral with both pairs of opposite sides parallel. Check the rhombus: It has two pairs of parallel sides, so it IS a parallelogram.
   Full step-by-step solution

   Step 1: Recall the definition of a parallelogram: a quadrilateral with both pairs of opposite sides parallel. Step 2: Check the rhombus: It has two pairs of parallel sides, so it IS a parallelogram. Step 3: Check the square: It has two pairs of parallel sides, so it IS a parallelogram. Step 4: Check the rectangle: It has two pairs of parallel sides, so it IS a parallelogram. Step 5: Check the trapezoid: By definition, a trapezoid has exactly one pair of parallel sides, so it is NOT a parallelogram. Step 6: Therefore, Isabella should move the rhombus, square, and rectangle into the 'Parallelogram' bin.

4. Aroha is sorting a collection of geometric tiles for an art project. She has a tile that is a quadrilateral with four right angles and all four sides of equal length. Aroha's friend Tane says, 'That tile is a rhombus, so it cannot be a square.' Is Tane correct? Explain why or why not using the properties of these shapes. Answer: No, Tane is not correct. A square is a special type of rhombus because it has all sides equal (like a rhombus) and also has four right angles. All squares are rhombuses, so the tile can be both a rhombus and a square. Tane's statement is wrong because being a rhombus does not prevent the shape from being a square—it actually includes it. Solution: Identify the properties of the tile. It has four sides (quadrilateral), four right angles, and all sides equal. Step 2: Define a rhombus: a quadrilateral with all four sides equal.
   Full step-by-step solution

   Step 1: Identify the properties of the tile. It has four sides (quadrilateral), four right angles, and all sides equal. Step 2: Define a rhombus: a quadrilateral with all four sides equal. The tile meets this condition, so it is a rhombus. Step 3: Define a square: a quadrilateral with all sides equal and four right angles. The tile also meets this condition, so it is a square. Step 4: Understand the hierarchy: Every square is a rhombus (because it has equal sides), but not every rhombus is a square (because a rhombus may not have right angles). Since the tile has both properties, it belongs to both categories. Therefore, Tane is incorrect: the tile can be a rhombus and a square at the same time.