2D Figure Classification

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Answer Key & Explanations

2D Figure Classification · Grade 5 · Worksheet 1

1. 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.

2. Sophia is helping her younger brother organize a set of quadrilateral cards for a school project. The set includes a square with sides of 506 mm, a rectangle that is 511 mm long and 506 mm wide, a rhombus with all sides of 506 mm but no right angles, and a parallelogram with sides of 511 mm and 506 mm and no right angles. Sophia says, 'The rectangle is a parallelogram, so it must also be a rhombus.' Her brother Noah disagrees. Who is correct, Sophia or Noah? Explain your reasoning using the properties of these shapes. Answer: Noah is correct. Solution: Recall the properties of each shape. A parallelogram has both pairs of opposite sides parallel. It has side lengths of 511 mm and 506 mm, so its opposite sides are equal, but all four sides are NOT equal (511 mm is not equal to 506 mm).
   Full step-by-step solution

   Step 1: Recall the properties of each shape. A parallelogram has both pairs of opposite sides parallel. A rectangle is a special type of parallelogram that also has four right angles. A rhombus is a special type of parallelogram that also has all four sides equal. Step 2: Look at the rectangle in the set. It has side lengths of 511 mm and 506 mm, so its opposite sides are equal, but all four sides are NOT equal (511 mm is not equal to 506 mm). Step 3: For a shape to be a rhombus, it must have all four sides equal. Since the rectangle has sides of two different lengths, it does NOT have all sides equal, so it cannot be a rhombus. Step 4: Sophia's statement is that the rectangle is a parallelogram (which is true) and therefore it must also be a rhombus (which is false). Noah disagrees, and he is correct because the rectangle does not meet the requirement of having all sides equal. Step 5: Final answer: Noah is correct.

3. Mere and Hana are designing a quilt for a school project. They have four different quadrilateral patches: a square with side length 502 mm, a rectangle that is 524 mm long and 502 mm wide, a rhombus with all sides 502 mm but no right angles, and a parallelogram with sides 524 mm and 502 mm and no right angles. Matiu says, 'The square is a rectangle and also a rhombus, but the rhombus is not a rectangle.' Hana says, 'The rectangle is a parallelogram, but the parallelogram is not a rectangle.' Mere says, 'The rhombus is a parallelogram, but the parallelogram is not a rhombus.' Who is correct? Explain your reasoning using the properties of each shape. Answer: All three are correct. Solution: Analyze Matiu's statement. The square has four equal sides (502 mm) and four right angles. A rectangle only needs four right angles, so the square qualifies.
   Full step-by-step solution

   Step 1: Analyze Matiu's statement. The square has four equal sides (502 mm) and four right angles. A rectangle only needs four right angles, so the square qualifies. A rhombus only needs four equal sides, so the square qualifies. The rhombus has four equal sides but no right angles, so it does not have four right angles, so it is not a rectangle. Matiu is correct. Step 2: Analyze Hana's statement. The rectangle has four right angles, so both pairs of opposite sides are parallel, which makes it a parallelogram. The parallelogram has opposite sides parallel but no right angles, so it does not have four right angles, so it is not a rectangle. Hana is correct. Step 3: Analyze Mere's statement. The rhombus has four equal sides, so both pairs of opposite sides are parallel, which makes it a parallelogram. The parallelogram has opposite sides parallel but its sides are not all equal (524 mm and 502 mm), so it is not a rhombus. Mere is correct. Step 4: Since all three statements are true, the answer is that all three are correct.