2D Figure Classification

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Answer Key & Explanations

2D Figure Classification · Grade 5 · Worksheet 3

1. Charlotte has drawn a large quadrilateral with 4 right angles and 4 sides of equal length. She then draws another quadrilateral that has 4 right angles but only opposite sides are equal in length. Mason says, 'Both of these shapes are rectangles, but only one is a square.' Is Mason correct? Explain your reasoning using the properties of these shapes. Answer: Yes, Mason is correct. A square is a special type of rectangle because it has all the properties of a rectangle (4 right angles, opposite sides parallel and equal) plus the additional property that all four sides are equal. So the first shape (with all sides equal) is a square, and it is also a rectangle. The second shape (with only opposite sides equal) is a rectangle but not a square. Solution: List the properties of a rectangle: 4 right angles, opposite sides are parallel, opposite sides are equal in length.
   Full step-by-step solution

   Step 1: List the properties of a rectangle: 4 right angles, opposite sides are parallel, opposite sides are equal in length. Step 2: List the properties of a square: 4 right angles, opposite sides are parallel, all 4 sides are equal in length. Step 3: Compare: A square has all the properties of a rectangle (4 right angles, parallel opposite sides, equal opposite sides), plus one more property (all sides equal). Step 4: Therefore, every square is a rectangle, but not every rectangle is a square. Step 5: Charlotte's first shape has 4 right angles and all sides equal — it is a square and a rectangle. Step 6: Charlotte's second shape has 4 right angles but only opposite sides equal — it is a rectangle but not a square. Step 7: Mason 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. Charlotte is designing a mosaic for her art class. She draws a quadrilateral with four equal sides and four right angles. Her friend Mason says, 'That shape is a square.' Her teacher Isabella says, 'But it is also a rectangle and a parallelogram.' Charlotte is confused. Why can the same shape be called a square, a rectangle, and a parallelogram? In your explanation, name the properties that make this shape fit all three categories. Answer: A square has four equal sides and four right angles, which makes it a rectangle (a parallelogram with four right angles) and a parallelogram (a quadrilateral with opposite sides parallel). Solution: Identify the properties of the shape. Charlotte's shape has four equal sides and four right angles. This is the definition of a square.
   Full step-by-step solution

   Step 1: Identify the properties of the shape. Charlotte's shape has four equal sides and four right angles. This is the definition of a square. Step 2: Check if it meets the definition of a rectangle. A rectangle is any quadrilateral with four right angles (and opposite sides equal). Since the square has four right angles, it is a special kind of rectangle. Step 3: Check if it meets the definition of a parallelogram. A parallelogram is any quadrilateral with both pairs of opposite sides parallel. A rectangle (and therefore a square) has opposite sides parallel, so the square is also a special kind of parallelogram. Step 4: Explain the hierarchy. A square is always a rectangle because it has four right angles. A square is always a parallelogram because its opposite sides are parallel. However, not all rectangles are squares (rectangles don't need equal sides), and not all parallelograms are rectangles (parallelograms don't need right angles). The square is the most specific shape in this group. The answer: The shape fits all three names because it meets the specific property requirements of each category: a square is a specialized type of rectangle and a specialized type of parallelogram.

4. Emma is sorting a collection of 2D geometric figures for a school project. She has the following shapes: a square with sides of 511 mm, a rectangle that is 523 mm long and 517 mm wide, a rhombus with all sides of 509 mm but no right angles, and a parallelogram with sides of 527 mm and 533 mm and no right angles. Emma says, 'Since my square has all sides equal and all right angles, it is the only shape in my collection that is also a rhombus.' Is Emma's statement correct? Explain your reasoning. Answer: No Solution: A rhombus is defined as a quadrilateral with all four sides of equal length. The square (511 mm sides) is a special type of rhombus because it has equal sides and right angles.
   Full step-by-step solution

   A rhombus is defined as a quadrilateral with all four sides of equal length. The square (511 mm sides) is a special type of rhombus because it has equal sides and right angles. However, the rectangle has two pairs of equal sides (523 mm and 517 mm), but not all four sides are equal, so it is not a rhombus. The rhombus in the collection (509 mm sides) is already a rhombus. The parallelogram has sides of two different lengths (527 mm and 533 mm), so it is not a rhombus either. Emma's statement is wrong because she thinks only the square is a rhombus, but the rhombus in her collection is also a rhombus, and she missed that. The square is one type of rhombus, but the other rhombus is also a rhombus.