Draw Shapes

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Answer Key & Explanations

Draw Shapes · Grade 7 · Worksheet 3

1. Draw a triangle with sides 13 cm, 14 cm, and 15 cm. What is the perimeter of this triangle in centimeters? Answer: 42 Solution: Identify the side lengths: 13 cm, 14 cm, and 15 cm. Add the three side lengths: 13 + 14 + 15 = 27 + 15 = 42. The perimeter is 42 cm.
   Full step-by-step solution

   Step 1: Identify the side lengths: 13 cm, 14 cm, and 15 cm. Step 2: Add the three side lengths: 13 + 14 + 15 = 27 + 15 = 42. Step 3: The perimeter is 42 cm. The answer is 42.

2. Draw a quadrilateral with sides 8 cm, 12 cm, 8 cm, and 12 cm, and one right angle. Answer: A parallelogram with one right angle (a rectangle) with sides 8 cm and 12 cm. Solution: The given side lengths are 8 cm, 12 cm, 8 cm, and 12 cm. Opposite sides are equal, so the shape is a parallelogram.
   Full step-by-step solution

   Step 1: The given side lengths are 8 cm, 12 cm, 8 cm, and 12 cm. Opposite sides are equal, so the shape is a parallelogram. Step 2: In a parallelogram, if one angle is 90 degrees, then all angles must be 90 degrees because consecutive angles are supplementary (sum to 180 degrees). Step 3: A parallelogram with all right angles is a rectangle. Step 4: Therefore, the shape is a rectangle with side lengths 8 cm and 12 cm. Step 5: To draw it, use a ruler to draw a horizontal side of 12 cm. At each endpoint, use a protractor to draw perpendicular lines of length 8 cm. Connect the top endpoints with a 12 cm line. The final shape is a rectangle. The answer is a rectangle with sides 8 cm and 12 cm.

3. A rectangular garden has a length of 12500 cm and a width of 8000 cm. The owner wants to divide it into square plots of the largest possible equal size without any leftover space. What is the side length, in centimeters, of each square plot? Answer: 500 Solution: We are given a rectangular garden with length = 12500 cm and width = 8000 cm. We want to divide it into equal square plots of the largest possible size without leftover space.
   Full step-by-step solution

   We are given a rectangular garden with length = 12500 cm and width = 8000 cm. We want to divide it into equal square plots of the largest possible size without leftover space. This means the side length of each square must divide both the length and the width evenly. So we need the greatest common divisor (GCD) of 12500 and 8000. --- **Step 1: Prime factorization of both numbers** First, factor 12500: 12500 = 125 × 100 125 = 5 × 25 = 5 × 5 × 5 = 5^3 100 = 4 × 25 = 2^2 × 5^2 So 12500 = 5^3 × (2^2 × 5^2) = 2^2 × 5^(3+2) = 2^2 × 5^5 So: 12500 = 2^2 × 5^5 --- Now factor 8000: 8000 = 8 × 1000 8 = 2^3 1000 = 10^3 = (2×5)^3 = 2^3 × 5^3 So 8000 = 2^3 × (2^3 × 5^3) = 2^(3+3) × 5^3 = 2^6 × 5^3 So: 8000 = 2^6 × 5^3 --- **Step 2: Find GCD using prime factors** GCD = take the smallest power of each common prime factor. Common primes: 2 and 5. For prime 2: 12500 has 2^2, 8000 has 2^6 → min exponent = 2 → 2^2 For prime 5: 12500 has 5^5, 8000 has 5^3 → min exponent = 3 → 5^3 So GCD = 2^2 × 5^3 --- **Step 3: Compute GCD value** 2^2 = 4 5^3 = 125 So GCD = 4 × 125 = 500 --- **Step 4: Interpretation** The largest possible square plot side length = GCD(12500, 8000) = 500 cm. Check: 12500 ÷ 500 = 25 squares along length 8000 ÷ 500 = 16 squares along width Total squares = 25 × 16 = 400 squares, no leftover space. --- **Final answer:** 500

4. Isabella is designing a parallelogram-shaped garden for her school's landscaping project. She wants the garden to have one side of length 15 meters, an adjacent side of length 10 meters, and one angle that measures 60 degrees. She must also ensure that the longer diagonal of the parallelogram is exactly 18 meters. Can a parallelogram with these conditions be drawn? Explain why or why not using the triangle inequality theorem and the Law of Cosines. Answer: No, a parallelogram with these conditions cannot be drawn because the given diagonal length of 18 meters violates the triangle inequality theorem. Solution: In a parallelogram, the longer diagonal connects opposite vertices. Consider the triangle formed by two adjacent sides (15 m and 10 m) and the angle between them (60 degrees).
   Full step-by-step solution

   Step 1: In a parallelogram, the longer diagonal connects opposite vertices. Consider the triangle formed by two adjacent sides (15 m and 10 m) and the angle between them (60 degrees). The diagonal opposite the 60-degree angle is the third side of this triangle. Step 2: Use the Law of Cosines to find the length of the diagonal (d) from the given sides and included angle: d^2 = 15^2 + 10^2 - 2(15)(10)cos(60) d^2 = 225 + 100 - 2(150)(0.5) d^2 = 325 - 150 d^2 = 175 d = sqrt(175) ≈ 13.23 meters Step 3: The diagonal we calculated is approximately 13.23 meters, but Isabella wants the diagonal to be 18 meters. Now check if a triangle with sides 15 m, 10 m, and 18 m is possible using the triangle inequality theorem: - 15 + 10 > 18? 25 > 18 (true) - 15 + 18 > 10? 33 > 10 (true) - 10 + 18 > 15? 28 > 15 (true) All three inequalities hold, so a triangle with sides 15, 10, and 18 is possible. Step 4: However, for a parallelogram, the diagonal must be opposite the given angle. The Law of Cosines shows that with sides 15 and 10 and included angle 60 degrees, the diagonal must be sqrt(175) ≈ 13.23 m. To have a diagonal of 18 m with the same sides, the included angle would need to be different (larger than 60 degrees). So the given conditions (side lengths, one angle, and diagonal length) are inconsistent. No parallelogram can satisfy all these conditions simultaneously.

5. Mason is drawing a quadrilateral with four sides of lengths 12 cm, 17 cm, 22 cm, and 17 cm. The quadrilateral has exactly one pair of parallel sides, and the two non-parallel sides are equal in length. Draw and identify the specific type of quadrilateral Mason is drawing. What is the name of this quadrilateral? Answer: isosceles trapezoid Solution: Identify the shape as a quadrilateral with one pair of parallel sides. By definition, a quadrilateral with exactly one pair of parallel sides is a trapezoid. The two non-parallel sides are both 17 cm, so they are equal in length.
   Full step-by-step solution

   Step 1: Identify the shape as a quadrilateral with one pair of parallel sides. By definition, a quadrilateral with exactly one pair of parallel sides is a trapezoid. Step 2: The two non-parallel sides are both 17 cm, so they are equal in length. Step 3: A trapezoid with equal non-parallel sides is called an isosceles trapezoid. The answer is isosceles trapezoid.