Draw Shapes

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Answer Key & Explanations

Draw Shapes · Grade 7 · Worksheet 1

1. Noah is drawing a quadrilateral on a coordinate grid. The vertices are located at A(7, 1), B(13, 1), C(16, 10), and D(10, 10). Draw this quadrilateral and determine its shape. What is the area of this quadrilateral? Answer: 54 square units Solution: Plot the points: A(7,1), B(13,1), C(16,10), D(10,10). Connect in order: A to B, B to C, C to D, D to A. Identify the shape.
   Full step-by-step solution

   Step 1: Plot the points: A(7,1), B(13,1), C(16,10), D(10,10). Connect in order: A to B, B to C, C to D, D to A. Step 2: Identify the shape. AB is horizontal from x=7 to x=13, so length = 13 - 7 = 6 units. DC is horizontal from x=10 to x=16, so length = 16 - 10 = 6 units. So AB and DC are parallel and equal. AD goes from (7,1) to (10,10): horizontal change = 3, vertical change = 9. BC goes from (13,1) to (16,10): horizontal change = 3, vertical change = 9. So AD and BC are parallel and equal. This is a parallelogram. Step 3: Base = length of AB = 6 units. Step 4: Height is the perpendicular distance between the horizontal sides AB and DC. AB is at y=1, DC is at y=10. Height = 10 - 1 = 9 units. Step 5: Area of parallelogram = base × height = 6 × 9 = 54 square units. The answer is 54 square units.

2. Hana is designing a triangular flag for her school's sports day. She has three wooden rods to use as the sides of the flag. The rods measure 24 centimeters, 30 centimeters, and 36 centimeters. Hana wants to draw the triangle on a large piece of paper first to plan the design. She knows she can draw a triangle if the side lengths satisfy a specific rule. Based on the side lengths, can Hana draw a triangle? Explain why or why not. Answer: Yes, because the sum of the lengths of any two sides is greater than the length of the third side (Triangle Inequality Theorem). Solution: Identify the three side lengths: 24 cm, 30 cm, and 36 cm. Find the longest side: 36 cm. Add the lengths of the two shorter sides: 24 cm + 30 cm = 54 cm.
   Full step-by-step solution

   Step 1: Identify the three side lengths: 24 cm, 30 cm, and 36 cm. Step 2: Find the longest side: 36 cm. Step 3: Add the lengths of the two shorter sides: 24 cm + 30 cm = 54 cm. Step 4: Compare the sum to the longest side: 54 cm > 36 cm. Step 5: Check the other two combinations to be thorough: - 24 cm + 36 cm = 60 cm > 30 cm (True) - 30 cm + 36 cm = 66 cm > 24 cm (True) Step 6: Since the sum of the lengths of any two sides is greater than the length of the third side, the Triangle Inequality Theorem is satisfied. Conclusion: Yes, Hana can draw a triangle with side lengths of 24 cm, 30 cm, and 36 cm.

3. Emma is designing a triangular garden with sides of 15 meters, 20 meters, and 25 meters. She wants to plant flowers along the entire perimeter and place a decorative fountain at a point that is equidistant from all three vertices of the triangular garden. How many meters is the fountain from each vertex of the garden? Answer: 12.5 Solution: The point equidistant from all three vertices of a triangle is called the circumcenter. For a triangle with sides 15 m, 20 m, and 25 m, we can check if it's a right triangle: 15^2 + 20^2 = 225 + 400 = 625, and 25^2 = 625.
   Full step-by-step solution

   Step 1: The point equidistant from all three vertices of a triangle is called the circumcenter. Step 2: For a triangle with sides 15 m, 20 m, and 25 m, we can check if it's a right triangle: 15^2 + 20^2 = 225 + 400 = 625, and 25^2 = 625. Since they're equal, this is a right triangle. Step 3: In a right triangle, the circumcenter is located at the midpoint of the hypotenuse. Step 4: The hypotenuse is 25 meters, so the midpoint is 25 ÷ 2 = 12.5 meters from each end. Step 5: Therefore, the fountain should be placed 12.5 meters from each vertex of the garden.

4. Emma is drawing a quadrilateral for an art project. She wants it to have one pair of parallel sides, with the longer parallel side measuring 13 cm and the shorter parallel side measuring 7 cm. The non-parallel sides are both 5 cm long. What type of quadrilateral is Emma drawing? Answer: isosceles trapezoid Solution: Identify the key conditions. The shape is a quadrilateral (4 sides). It has one pair of parallel sides (the sides measuring 13 cm and 7 cm are parallel).
   Full step-by-step solution

   Step 1: Identify the key conditions. The shape is a quadrilateral (4 sides). It has one pair of parallel sides (the sides measuring 13 cm and 7 cm are parallel). This makes it a trapezoid. Step 2: The other two sides (non-parallel) are both 5 cm long, meaning they are equal in length. Step 3: A trapezoid with equal non-parallel sides is called an isosceles trapezoid. Therefore, Emma is drawing an isosceles trapezoid.

5. Isabella is drawing a quadrilateral with two right angles. The first right angle is at vertex A, and the second right angle is at vertex C. The side lengths are: AB = 12 cm, BC = 16 cm, CD = 12 cm, and DA = 16 cm. What is the shape of this quadrilateral? Answer: rectangle Solution: The quadrilateral has vertices A, B, C, D in order. Right angles at A and C are opposite vertices. Side lengths are AB = 12 cm, BC = 16 cm, CD = 12 cm, DA = 16 cm.
   Full step-by-step solution

   Step 1: The quadrilateral has vertices A, B, C, D in order. Right angles at A and C are opposite vertices. Step 2: Side lengths are AB = 12 cm, BC = 16 cm, CD = 12 cm, DA = 16 cm. This means opposite sides are equal: AB = CD = 12 cm and BC = DA = 16 cm. Step 3: A quadrilateral with opposite sides equal is a parallelogram. With right angles at two opposite vertices, all angles become right angles because adjacent angles in a parallelogram are supplementary. Step 4: A parallelogram with all right angles is a rectangle. Since side lengths are 12 and 16, not equal, it is not a square. Step 5: The shape is a rectangle with dimensions 12 cm by 16 cm. The answer is rectangle.

6. Draw a triangle with side lengths 14 cm, 22 cm, and 28 cm. What is the perimeter of the triangle? Answer: 64 cm Solution: Identify the side lengths: 14 cm, 22 cm, and 28 cm. Add the side lengths: 14 + 22 = 36. Add the third side: 36 + 28 = 64.
   Full step-by-step solution

   Step 1: Identify the side lengths: 14 cm, 22 cm, and 28 cm. Step 2: Add the side lengths: 14 + 22 = 36. Step 3: Add the third side: 36 + 28 = 64. Step 4: Include the unit: 64 cm. The perimeter of the triangle is 64 cm.

7. Draw a triangle with side lengths 9 cm, 11 cm, and 15 cm. Answer: A scalene triangle with sides 9 cm, 11 cm, and 15 cm Solution: Check the triangle inequality theorem. - 9 + 11 = 20, which is greater than 15. ✓ - 9 + 15 = 24, which is greater than 11.
   Full step-by-step solution

   Step 1: Check the triangle inequality theorem. - 9 + 11 = 20, which is greater than 15. ✓ - 9 + 15 = 24, which is greater than 11. ✓ - 11 + 15 = 26, which is greater than 9. ✓ All conditions are satisfied, so a triangle can be drawn. Step 2: Draw the longest side (15 cm) as the base. Step 3: Use a compass or ruler to mark points 9 cm from one endpoint and 11 cm from the other endpoint. The intersection of these arcs gives the third vertex. Step 4: Connect the vertices to form the triangle. This triangle has no equal sides, so it is a scalene triangle. The answer is a scalene triangle with sides 9 cm, 11 cm, and 15 cm.

8. (3/4 + 2/3) × 12 = ? Answer: 17 Solution: Add the fractions inside the parentheses. We have 3/4 + 2/3. To add these, we need a common denominator.
   Full step-by-step solution

   Step 1: Add the fractions inside the parentheses. We have 3/4 + 2/3. To add these, we need a common denominator. The least common denominator of 4 and 3 is 12. Step 2: Convert each fraction to have denominator 12. 3/4 = (3 × 3)/(4 × 3) = 9/12 2/3 = (2 × 4)/(3 × 4) = 8/12 Step 3: Add the converted fractions. 9/12 + 8/12 = (9 + 8)/12 = 17/12 Step 4: Now the expression is (17/12) × 12. Multiplying a fraction by its denominator simplifies nicely: (17/12) × 12 = 17 × (12/12) = 17 × 1 = 17 Final Answer: 17