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Coordinate Polygons

Grade 6 · Geometry · Worksheet 2

  1. (-24 + 15) × 3 + 36 ÷ (-4) = ? Answer: ______________
  2. A triangle has vertices at A(-9, -8), B(11, -8), and C(-9, 10). What is the area of the triangle? Answer: ______________
  3. Liam is designing a rectangular garden for his school project. He plots the vertices on a coordinate plane at points A(2, 3), B(10, 3), C(10, 8), and D(2, 8). He wants to build a fence around the entire perimeter of the garden. What is the total length of fencing, in units, that Liam will need? Answer: ______________
  4. (-18 ÷ 3) + (5 × 4) = ? Answer: ______________
  5. Isabella is designing a triangular garden on a coordinate plane for her school's landscaping project. The vertices of the garden are at points A(-7, -2), B(7, -2), and C(1, 10). She needs to know the perimeter of the garden to buy enough fencing. What is the perimeter of Isabella's triangular garden, rounded to the nearest whole unit? Answer: ______________
  6. A triangle has vertices at (-5, -3), (3, -3), and (1, 5). What is the area of the triangle? Answer: ______________
  7. A quadrilateral has vertices at (-12, 15), (18, 15), (18, -10), and (-12, -10). What is the area of this quadrilateral? Answer: ______________
  8. A rectangular garden is plotted on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). A diagonal path is drawn from (2, 1) to (8, 5), dividing the garden into two triangular sections. What is the area of one of these triangular sections? Answer: ______________
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Answer Key & Explanations

Coordinate Polygons · Grade 6 · Worksheet 2

  1. (-24 + 15) × 3 + 36 ÷ (-4) = ? Answer: -36 Solution: Calculate inside the parentheses: (-24 + 15) = -9 Perform multiplication: (-9) × 3 = -27 Perform division: 36 ÷ (-4) = -9 Add the results: -27 + (-9) = -36 The answer is -36.
    Full step-by-step solution

    Step 1: Calculate inside the parentheses: (-24 + 15) = -9 Step 2: Perform multiplication: (-9) × 3 = -27 Step 3: Perform division: 36 ÷ (-4) = -9 Step 4: Add the results: -27 + (-9) = -36 The answer is -36.

  2. A triangle has vertices at A(-9, -8), B(11, -8), and C(-9, 10). What is the area of the triangle? Answer: 180 Solution: Identify the base. Points A(-9, -8) and B(11, -8) have the same y-coordinate (-8), so AB is horizontal. The length of AB is the difference in x-coordinates: 11 - (-9) = 20 units.
    Full step-by-step solution

    Step 1: Identify the base. Points A(-9, -8) and B(11, -8) have the same y-coordinate (-8), so AB is horizontal. The length of AB is the difference in x-coordinates: 11 - (-9) = 20 units. Step 2: Identify the height. Points A(-9, -8) and C(-9, 10) have the same x-coordinate (-9), so AC is vertical. The length of AC is the difference in y-coordinates: 10 - (-8) = 18 units. Step 3: The triangle is right-angled at A, with base AB = 20 and height AC = 18. Step 4: Area of a triangle = (1/2) × base × height = (1/2) × 20 × 18 = 10 × 18 = 180 square units. The answer is 180.

  3. Liam is designing a rectangular garden for his school project. He plots the vertices on a coordinate plane at points A(2, 3), B(10, 3), C(10, 8), and D(2, 8). He wants to build a fence around the entire perimeter of the garden. What is the total length of fencing, in units, that Liam will need? Answer: 26 Solution: A(2, 3) B(10, 3) C(10, 8) D(2, 8) A to B: horizontal line at y = 3 B to C: vertical line at x = 10 C to D: horizontal line at y = 8 D to A: vertical line at x = 2 This is a rectangle.
    Full step-by-step solution

    Let's solve this step by step. **Step 1: Identify the shape and coordinates** The vertices are: A(2, 3) B(10, 3) C(10, 8) D(2, 8) Plotting these points mentally, we see: A to B: horizontal line at y = 3 B to C: vertical line at x = 10 C to D: horizontal line at y = 8 D to A: vertical line at x = 2 This is a rectangle. **Step 2: Find the length of side AB** A(2, 3) and B(10, 3) have the same y-coordinate, so the length is the difference in x-coordinates: Length AB = 10 - 2 = 8 units. **Step 3: Find the length of side BC** B(10, 3) and C(10, 8) have the same x-coordinate, so the length is the difference in y-coordinates: Length BC = 8 - 3 = 5 units. **Step 4: Find the length of side CD** C(10, 8) and D(2, 8) have the same y-coordinate, so the length is the difference in x-coordinates: Length CD = 10 - 2 = 8 units. **Step 5: Find the length of side DA** D(2, 8) and A(2, 3) have the same x-coordinate, so the length is the difference in y-coordinates: Length DA = 8 - 3 = 5 units. **Step 6: Calculate the perimeter** Perimeter = AB + BC + CD + DA Perimeter = 8 + 5 + 8 + 5 Perimeter = 26 units. **Step 7: Conclusion** The total length of fencing Liam needs is 26 units.

  4. (-18 ÷ 3) + (5 × 4) = ? Answer: 14 Solution: Solve the operations inside the parentheses first. Start with (-18 ÷ 3). -18 ÷ 3 = -6.
    Full step-by-step solution

    Step 1: Solve the operations inside the parentheses first. Start with (-18 ÷ 3). Step 2: -18 ÷ 3 = -6. Step 3: Now solve (5 × 4). Step 4: 5 × 4 = 20. Step 5: The expression is now -6 + 20. Step 6: -6 + 20 = 14. The answer is 14.

  5. Isabella is designing a triangular garden on a coordinate plane for her school's landscaping project. The vertices of the garden are at points A(-7, -2), B(7, -2), and C(1, 10). She needs to know the perimeter of the garden to buy enough fencing. What is the perimeter of Isabella's triangular garden, rounded to the nearest whole unit? Answer: 42 Solution: Calculate the length of side AB. Points A(-7, -2) and B(7, -2) have the same y-coordinate (-2), so AB is horizontal. Length AB = 7 - (-7) = 7 + 7 = 14 units.
    Full step-by-step solution

    Step 1: Calculate the length of side AB. Points A(-7, -2) and B(7, -2) have the same y-coordinate (-2), so AB is horizontal. Length AB = 7 - (-7) = 7 + 7 = 14 units. Step 2: Calculate the length of side BC. Points B(7, -2) and C(1, 10) have different x and y coordinates, so BC is diagonal. Use the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) BC = sqrt((1 - 7)^2 + (10 - (-2))^2) BC = sqrt((-6)^2 + (12)^2) BC = sqrt(36 + 144) BC = sqrt(180) BC = sqrt(36 * 5) BC = 6 * sqrt(5) sqrt(5) ≈ 2.236, so BC ≈ 6 * 2.236 = 13.416 units. Step 3: Calculate the length of side CA. Points C(1, 10) and A(-7, -2) have different x and y coordinates, so CA is diagonal. CA = sqrt((-7 - 1)^2 + (-2 - 10)^2) CA = sqrt((-8)^2 + (-12)^2) CA = sqrt(64 + 144) CA = sqrt(208) CA = sqrt(16 * 13) CA = 4 * sqrt(13) sqrt(13) ≈ 3.606, so CA ≈ 4 * 3.606 = 14.424 units. Step 4: Find the perimeter. Perimeter = AB + BC + CA Perimeter = 14 + 13.416 + 14.424 Perimeter = 41.84 units. Step 5: Round to the nearest whole unit. 41.84 rounded to the nearest whole unit is 42. The perimeter of Isabella's triangular garden is 42 units.

  6. A triangle has vertices at (-5, -3), (3, -3), and (1, 5). What is the area of the triangle? Answer: 32 Solution: Identify the base. The vertices (-5, -3) and (3, -3) have the same y-coordinate (-3), so the side connecting them is horizontal. Base length = distance between x-coordinates: 3 - (-5) = 3 + 5 = 8 units.
    Full step-by-step solution

    Step 1: Identify the base. The vertices (-5, -3) and (3, -3) have the same y-coordinate (-3), so the side connecting them is horizontal. Base length = distance between x-coordinates: 3 - (-5) = 3 + 5 = 8 units. Step 2: Find the height. The third vertex is (1, 5). The height is the vertical distance from this vertex to the base line y = -3. Height = 5 - (-3) = 5 + 3 = 8 units. Step 3: Calculate the area. Area = 1/2 × base × height = 1/2 × 8 × 8 = 1/2 × 64 = 32 square units. The answer is 32.

  7. A quadrilateral has vertices at (-12, 15), (18, 15), (18, -10), and (-12, -10). What is the area of this quadrilateral? Answer: 750 Solution: Identify the shape. The vertices are (-12, 15), (18, 15), (18, -10), (-12, -10). This is a rectangle because opposite sides are parallel and all angles are right angles.
    Full step-by-step solution

    Step 1: Identify the shape. The vertices are (-12, 15), (18, 15), (18, -10), (-12, -10). This is a rectangle because opposite sides are parallel and all angles are right angles. Step 2: Find the width (horizontal distance). Use the top two points (-12, 15) and (18, 15). The width = 18 - (-12) = 18 + 12 = 30. Step 3: Find the height (vertical distance). Use the left side points (-12, 15) and (-12, -10). The height = 15 - (-10) = 15 + 10 = 25. Step 4: Calculate the area of the rectangle: Area = width × height = 30 × 25 = 750. The answer is 750.

  8. A rectangular garden is plotted on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). A diagonal path is drawn from (2, 1) to (8, 5), dividing the garden into two triangular sections. What is the area of one of these triangular sections? Answer: 12 Solution: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The diagonal is drawn from A(2, 1) to C(8, 5). This splits the rectangle into two congruent triangles: triangle ABC and triangle ADC.
    Full step-by-step solution

    Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangle with vertices: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The diagonal is drawn from A(2, 1) to C(8, 5). This splits the rectangle into two congruent triangles: triangle ABC and triangle ADC. We need the area of one of these triangles. --- **Step 2: Find the area of the rectangle** Length along x-axis: from x = 2 to x = 8 → length = 8 - 2 = 6 Width along y-axis: from y = 1 to y = 5 → width = 5 - 1 = 4 Area of rectangle = length × width = 6 × 4 = 24 --- **Step 3: Area of one triangle** A diagonal divides a rectangle into two triangles of equal area. So, area of one triangle = (Area of rectangle) / 2 = 24 / 2 = 12 --- **Step 4: Conclusion** The area of one triangular section is 12. --- **Final answer:** 12