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Coordinate Polygons

Grade 6 · Geometry · Worksheet 2

  1. A quadrilateral has vertices at (-8, 9), (11, 9), (11, -4), and (-8, -4). What is the perimeter of this quadrilateral? Answer: ______________
  2. A polygon has vertices at (-7, 2), (7, 2), (7, -7), and (-7, -7). What is the area of this polygon? 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. 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: ______________
  5. (-18 + 9) × 4 = ? Answer: ______________
  6. Matiu draws a quadrilateral on a coordinate plane with vertices at A(-8, 6), B(4, 6), C(4, -10), and D(-8, -10). He then draws a line segment from point A to point C, dividing the quadrilateral into two triangles. What is the area of triangle ABC? Answer: ______________
  7. Hana is creating a rectangular mosaic on a coordinate plane for her art project. The four corners of the mosaic are at points A(-8, -6), B(8, -6), C(8, 10), and D(-8, 10). She wants to glue a ribbon border around the entire perimeter of the rectangle. Each unit on the coordinate plane represents 1 centimeter. What is the total length of ribbon Hana needs, in centimeters? Answer: ______________
  8. Tane draws a quadrilateral on a coordinate plane with vertices at A(-21, 13), B(19, 13), C(19, -11), and D(-21, -11). He then draws a line segment from point B to point D, dividing the quadrilateral into two triangles. What is the area of triangle ABD? Answer: ______________
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Answer Key & Explanations

Coordinate Polygons · Grade 6 · Worksheet 2

  1. A quadrilateral has vertices at (-8, 9), (11, 9), (11, -4), and (-8, -4). What is the perimeter of this quadrilateral? Answer: 64 Solution: Identify the shape. The vertices are (-8,9), (11,9), (11,-4), (-8,-4). Points with same y-coordinate give horizontal sides: (-8,9) to (11,9) and (-8,-4) to (11,-4).
    Full step-by-step solution

    Step 1: Identify the shape. The vertices are (-8,9), (11,9), (11,-4), (-8,-4). Points with same y-coordinate give horizontal sides: (-8,9) to (11,9) and (-8,-4) to (11,-4). Points with same x-coordinate give vertical sides: (-8,9) to (-8,-4) and (11,9) to (11,-4). So it is a rectangle. Step 2: Find the horizontal side length. From x = -8 to x = 11, the distance is 11 - (-8) = 11 + 8 = 19 units. Step 3: Find the vertical side length. From y = 9 to y = -4, the distance is 9 - (-4) = 9 + 4 = 13 units. Step 4: Perimeter of a rectangle = 2 × (length + width) = 2 × (19 + 13) = 2 × 32 = 64 units. The answer is 64.

  2. A polygon has vertices at (-7, 2), (7, 2), (7, -7), and (-7, -7). What is the area of this polygon? Answer: 126 Solution: Plot the vertices: (-7, 2), (7, 2), (7, -7), (-7, -7). Connecting them in order forms a rectangle. Find the length of the horizontal side.
    Full step-by-step solution

    Step 1: Plot the vertices: (-7, 2), (7, 2), (7, -7), (-7, -7). Connecting them in order forms a rectangle. Step 2: Find the length of the horizontal side. The y-coordinates are both 2, so the length is the difference in x-coordinates: 7 - (-7) = 7 + 7 = 14. Step 3: Find the length of the vertical side. The x-coordinates are both 7, so the length is the difference in y-coordinates: 2 - (-7) = 2 + 7 = 9. Step 4: Area of a rectangle = length × width = 14 × 9 = 126. The answer is 126.

  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. 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.

  5. (-18 + 9) × 4 = ? Answer: -36 Solution: Start with the expression (-18 + 9) × 4 Calculate inside the parentheses first: -18 + 9 = -9 Now multiply the result by 4: -9 × 4 = -36 The final answer is -36.
    Full step-by-step solution

    Step 1: Start with the expression (-18 + 9) × 4 Step 2: Calculate inside the parentheses first: -18 + 9 = -9 Step 3: Now multiply the result by 4: -9 × 4 = -36 Step 4: The final answer is -36.

  6. Matiu draws a quadrilateral on a coordinate plane with vertices at A(-8, 6), B(4, 6), C(4, -10), and D(-8, -10). He then draws a line segment from point A to point C, dividing the quadrilateral into two triangles. What is the area of triangle ABC? Answer: 96 Solution: Plot the points: A(-8, 6), B(4, 6), C(4, -10), D(-8, -10). The quadrilateral is a rectangle because opposite sides are parallel and equal. Triangle ABC has vertices A(-8, 6), B(4, 6), and C(4, -10).
    Full step-by-step solution

    Step 1: Plot the points: A(-8, 6), B(4, 6), C(4, -10), D(-8, -10). The quadrilateral is a rectangle because opposite sides are parallel and equal. Step 2: Triangle ABC has vertices A(-8, 6), B(4, 6), and C(4, -10). Step 3: Find the base of triangle ABC. Side AB is horizontal from x = -8 to x = 4. Length AB = 4 - (-8) = 12 units. Step 4: Find the height of triangle ABC. The height is the vertical distance from point C down to line AB. Point C has y = -10, and line AB has y = 6. Height = 6 - (-10) = 16 units. Step 5: Area of triangle = (1/2) * base * height = (1/2) * 12 * 16 = 6 * 16 = 96 square units. The answer is 96.

  7. Hana is creating a rectangular mosaic on a coordinate plane for her art project. The four corners of the mosaic are at points A(-8, -6), B(8, -6), C(8, 10), and D(-8, 10). She wants to glue a ribbon border around the entire perimeter of the rectangle. Each unit on the coordinate plane represents 1 centimeter. What is the total length of ribbon Hana needs, in centimeters? Answer: 64 Solution: Find the length of the horizontal sides. Side AB: A(-8, -6) to B(8, -6). Both points have y = -6, so this side is horizontal.
    Full step-by-step solution

    Step 1: Find the length of the horizontal sides. Side AB: A(-8, -6) to B(8, -6). Both points have y = -6, so this side is horizontal. Length = 8 - (-8) = 8 + 8 = 16 centimeters. Side CD: C(8, 10) to D(-8, 10). Both points have y = 10, so this side is also horizontal. Length = 8 - (-8) = 8 + 8 = 16 centimeters. Step 2: Find the length of the vertical sides. Side BC: B(8, -6) to C(8, 10). Both points have x = 8, so this side is vertical. Length = 10 - (-6) = 10 + 6 = 16 centimeters. Side DA: D(-8, 10) to A(-8, -6). Both points have x = -8, so this side is also vertical. Length = 10 - (-6) = 10 + 6 = 16 centimeters. Step 3: Add all four side lengths to find the perimeter (total ribbon needed). Perimeter = 16 + 16 + 16 + 16 = 64 centimeters. Final answer: 64 centimeters.

  8. Tane draws a quadrilateral on a coordinate plane with vertices at A(-21, 13), B(19, 13), C(19, -11), and D(-21, -11). He then draws a line segment from point B to point D, dividing the quadrilateral into two triangles. What is the area of triangle ABD? Answer: 480 Solution: Plot the points: A(-21, 13), B(19, 13), C(19, -11), D(-21, -11). The x-coordinates are -21 and 19, and the y-coordinates are 13 and -11, so the quadrilateral is a rectangle.
    Full step-by-step solution

    Step 1: Plot the points: A(-21, 13), B(19, 13), C(19, -11), D(-21, -11). The x-coordinates are -21 and 19, and the y-coordinates are 13 and -11, so the quadrilateral is a rectangle. Step 2: Triangle ABD has vertices A(-21, 13), B(19, 13), and D(-21, -11). Step 3: Side AB is horizontal from x = -21 to x = 19. Length AB = 19 - (-21) = 19 + 21 = 40 units. Step 4: The height of triangle ABD is the vertical distance from point D to line AB. Point D has y = -11, and line AB has y = 13. Height = 13 - (-11) = 13 + 11 = 24 units. Step 5: Area of triangle = 1/2 x base x height = 1/2 x 40 x 24 = 20 x 24 = 480 square units. The answer is 480.