Coordinate Polygons

lessonbunny.com

Answer Key & Explanations

Coordinate Polygons · Grade 6 · Worksheet 3

1. Hana draws a quadrilateral on a coordinate plane with vertices at (-12, 15), (18, 15), (18, -9), and (-12, -9). What is the area of this quadrilateral? Answer: 720 Solution: Identify the shape. The vertices are (-12, 15), (18, 15), (18, -9), and (-12, -9). Notice that the x-coordinates are -12 and 18, and the y-coordinates are 15 and -9.
   Full step-by-step solution

   Step 1: Identify the shape. The vertices are (-12, 15), (18, 15), (18, -9), and (-12, -9). Notice that the x-coordinates are -12 and 18, and the y-coordinates are 15 and -9. This means the sides are horizontal and vertical, so the quadrilateral is a rectangle. Step 2: Find the length. The length is the horizontal distance between x = -12 and x = 18. Distance = 18 - (-12) = 18 + 12 = 30 units. Step 3: Find the width. The width is the vertical distance between y = 15 and y = -9. Distance = 15 - (-9) = 15 + 9 = 24 units. Step 4: Calculate the area. Area of a rectangle = length × width = 30 × 24 = 720 square units. The answer is 720.

2. A quadrilateral has vertices at A(-6, 1), B(1, 1), C(1, -6), and D(-6, -6). What is the perimeter of this quadrilateral? Answer: 28 Solution: Identify the shape. The vertices are A(-6,1), B(1,1), C(1,-6), D(-6,-6). Since A and B share y=1, AB is horizontal.
   Full step-by-step solution

   Step 1: Identify the shape. The vertices are A(-6,1), B(1,1), C(1,-6), D(-6,-6). Since A and B share y=1, AB is horizontal. B and C share x=1, BC is vertical. C and D share y=-6, CD is horizontal. D and A share x=-6, DA is vertical. So it is a rectangle. Step 2: Find side lengths. AB: from x=-6 to x=1, length = 1 - (-6) = 7 units. BC: from y=1 to y=-6, length = 1 - (-6) = 7 units. CD: from x=1 to x=-6, length = 1 - (-6) = 7 units. DA: from y=-6 to y=1, length = 1 - (-6) = 7 units. Step 3: Calculate perimeter. Perimeter = 7 + 7 + 7 + 7 = 28 units. The answer is 28.

3. (-18 + 12) × 4 - 24 ÷ (-3) = ? Answer: -16 Solution: Calculate inside the parentheses: (-18 + 12) = -6 Perform multiplication: (-6) × 4 = -24 Perform division: 24 ÷ (-3) = -8 Substitute back into the expression: -24 - (-8) Simplify the subtraction of a negative: -24 + 8 = -16 The answer is -16.
   Full step-by-step solution

   Step 1: Calculate inside the parentheses: (-18 + 12) = -6 Step 2: Perform multiplication: (-6) × 4 = -24 Step 3: Perform division: 24 ÷ (-3) = -8 Step 4: Substitute back into the expression: -24 - (-8) Step 5: Simplify the subtraction of a negative: -24 + 8 = -16 The answer is -16.

4. (-24 + 16) × 5 + 36 ÷ (-4) = ? Answer: -49 Solution: Calculate inside the parentheses: -24 + 16 = -8 Perform multiplication: -8 × 5 = -40 Perform division: 36 ÷ (-4) = -9 Add the results: -40 + (-9) = -49 The answer is -49.
   Full step-by-step solution

   Step 1: Calculate inside the parentheses: -24 + 16 = -8 Step 2: Perform multiplication: -8 × 5 = -40 Step 3: Perform division: 36 ÷ (-4) = -9 Step 4: Add the results: -40 + (-9) = -49 The answer is -49.

5. A city park is designed on a coordinate plane with a rectangular field from (0,0) to (25,0) to (25,18) to (0,18). A triangular pond is created inside the field with vertices at (0,0), (25,0), and (12.5,9). What percentage of the total park area is occupied by the triangular pond? Answer: 25 Solution: Length = 25 - 0 = 25 units Width = 18 - 0 = 18 units Area of rectangle = length × width = 25 × 18 = 450 square units Base = 25 - 0 = 25 units Height = 9 - 0 = 9 units Area of triangle = (1/2) × base × height = (1/2) × 25 × 9 = (1/2) × 225 = 112.5 square units Percentage = (area of triangle ÷…
   Full step-by-step solution

   Step 1: Calculate the area of the rectangular field Length = 25 - 0 = 25 units Width = 18 - 0 = 18 units Area of rectangle = length × width = 25 × 18 = 450 square units Step 2: Calculate the area of the triangular pond Base = 25 - 0 = 25 units Height = 9 - 0 = 9 units Area of triangle = (1/2) × base × height = (1/2) × 25 × 9 = (1/2) × 225 = 112.5 square units Step 3: Calculate the percentage Percentage = (area of triangle ÷ area of rectangle) × 100 Percentage = (112.5 ÷ 450) × 100 = 0.25 × 100 = 25% The answer is 25.

6. Maria is designing a triangular park on a coordinate plane. The park's vertices are at points A(-3, 2), B(5, 2), and C(1, 8). She wants to build a fence around the entire perimeter of the park. What is the total length of fencing Maria needs, rounded to the nearest tenth of a unit? Answer: 22.4 Solution: Calculate the distance between points A(-3, 2) and B(5, 2) using the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) AB = sqrt((5 - (-3))^2 + (2 - 2)^2) = sqrt((8)^2 + (0)^2) = sqrt(64 + 0) = sqrt(64) = 8 Calculate the distance between points B(5, 2) and C(1, 8) BC = sqrt((1 - 5)^2…
   Full step-by-step solution

   Step 1: Calculate the distance between points A(-3, 2) and B(5, 2) using the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) AB = sqrt((5 - (-3))^2 + (2 - 2)^2) = sqrt((8)^2 + (0)^2) = sqrt(64 + 0) = sqrt(64) = 8 Step 2: Calculate the distance between points B(5, 2) and C(1, 8) BC = sqrt((1 - 5)^2 + (8 - 2)^2) = sqrt((-4)^2 + (6)^2) = sqrt(16 + 36) = sqrt(52) ≈ 7.211 Step 3: Calculate the distance between points C(1, 8) and A(-3, 2) CA = sqrt((-3 - 1)^2 + (2 - 8)^2) = sqrt((-4)^2 + (-6)^2) = sqrt(16 + 36) = sqrt(52) ≈ 7.211 Step 4: Add all three sides to find the perimeter Perimeter = AB + BC + CA = 8 + 7.211 + 7.211 = 22.422 Step 5: Round to the nearest tenth 22.422 rounded to the nearest tenth is 22.4 The total length of fencing Maria needs is 22.4 units.

7. A triangle has vertices at A(-6, -1), B(6, -1), and C(1, 6). What is the area of triangle ABC? Answer: 42 Solution: Identify the base. Points A(-6, -1) and B(6, -1) have the same y-coordinate (-1), so AB is a horizontal segment. The length of AB is the difference in x-coordinates: 6 - (-6) = 12 units.
   Full step-by-step solution

   Step 1: Identify the base. Points A(-6, -1) and B(6, -1) have the same y-coordinate (-1), so AB is a horizontal segment. The length of AB is the difference in x-coordinates: 6 - (-6) = 12 units. So base = 12. Step 2: Find the height. The height is the vertical distance from point C(1, 6) to the line containing AB (y = -1). The height is the difference in y-coordinates: 6 - (-1) = 7 units. Step 3: Use the area formula for a triangle: Area = 1/2 × base × height = 1/2 × 12 × 7 = 6 × 7 = 42. The area of triangle ABC is 42 square units.

8. (-15 + 9) × 4 - 18 ÷ (-3) = ? Answer: -18 Solution: Calculate inside the parentheses: -15 + 9 = -6 Multiply: -6 × 4 = -24 Divide: 18 ÷ (-3) = -6 Subtract: -24 - (-6) = -24 + 6 = -18 The answer is -18.
   Full step-by-step solution

   Step 1: Calculate inside the parentheses: -15 + 9 = -6 Step 2: Multiply: -6 × 4 = -24 Step 3: Divide: 18 ÷ (-3) = -6 Step 4: Subtract: -24 - (-6) = -24 + 6 = -18 The answer is -18.

9. A triangle has vertices at (-9, 8), (11, 8), and (-9, -10). What is the area of the triangle? Answer: 180 Solution: Identify the base. The points (-9, 8) and (11, 8) share the same y-coordinate (8), so the base is horizontal. Length of base = 11 - (-9) = 11 + 9 = 20 units.
   Full step-by-step solution

   Step 1: Identify the base. The points (-9, 8) and (11, 8) share the same y-coordinate (8), so the base is horizontal. Length of base = 11 - (-9) = 11 + 9 = 20 units. Step 2: Identify the height. The points (-9, 8) and (-9, -10) share the same x-coordinate (-9), so the height is vertical. Length of height = 8 - (-10) = 8 + 10 = 18 units. Step 3: Area of a triangle = (1/2) × base × height = (1/2) × 20 × 18 = 10 × 18 = 180 square units. The answer is 180.