Coordinate Polygons

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Answer Key & Explanations

Coordinate Polygons · Grade 6 · Worksheet 1

1. Mason designs a kite on a coordinate plane. The vertices of the kite are at (-7, 2), (0, 12), (7, 2), and (0, -8). What is the area of the kite in square units? Answer: 140 Solution: Identify the diagonals. The kite has vertices A(-7, 2), B(0, 12), C(7, 2), and D(0, -8). The diagonals are AC (from left to right) and BD (from top to bottom).
   Full step-by-step solution

   Step 1: Identify the diagonals. The kite has vertices A(-7, 2), B(0, 12), C(7, 2), and D(0, -8). The diagonals are AC (from left to right) and BD (from top to bottom). Step 2: Find the length of diagonal AC. A is at (-7, 2) and C is at (7, 2). Since they have the same y-coordinate, the length is the difference in x-coordinates: 7 - (-7) = 14 units. Step 3: Find the length of diagonal BD. B is at (0, 12) and D is at (0, -8). Since they have the same x-coordinate, the length is the difference in y-coordinates: 12 - (-8) = 20 units. Step 4: Area of a kite = (product of diagonals) / 2 = (14 * 20) / 2 = 280 / 2 = 140 square units. The answer is 140.

2. A quadrilateral has vertices at (-5, 0), (0, 5), (5, 0), and (0, -5). What is the area of this quadrilateral? Answer: 50 Solution: Plot the points: (-5, 0), (0, 5), (5, 0), (0, -5). This is a rhombus (actually a square rotated 45 degrees).
   Full step-by-step solution

   Step 1: Plot the points: (-5, 0), (0, 5), (5, 0), (0, -5). This is a rhombus (actually a square rotated 45 degrees). Step 2: The diagonals of this quadrilateral are from (-5, 0) to (5, 0) (horizontal diagonal) and from (0, -5) to (0, 5) (vertical diagonal). Step 3: Length of horizontal diagonal = 5 - (-5) = 10 units. Step 4: Length of vertical diagonal = 5 - (-5) = 10 units. Step 5: The area of a rhombus (or kite) can be found using the formula: Area = (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals. Step 6: Area = (10 × 10) / 2 = 100 / 2 = 50 square units. The answer is 50.

3. Liam is designing a rectangular garden on a coordinate plane. The garden's corners are at points A(2, 5), B(8, 5), C(8, 12), and D(2, 12). He wants to build a fence around the entire perimeter and then install a diagonal stone path from point A to point C. What is the total length of fencing Liam needs, and how long will the stone path be? Answer: 26 units and 13 units Solution: When working with polygons on a coordinate plane, you can use the coordinates to find side lengths. For a rectangle, opposite sides are equal.
   Full step-by-step solution

   When working with polygons on a coordinate plane, you can use the coordinates to find side lengths. For a rectangle, opposite sides are equal. The diagonal of a rectangle creates two congruent right triangles, allowing you to use the Pythagorean theorem to find its length.

4. (-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.

5. A quadrilateral has vertices at (-6, 4), (2, 4), (2, -4), and (-6, -4). What is the area of this quadrilateral? Answer: 64 Solution: Plot the points: (-6, 4), (2, 4), (2, -4), (-6, -4). Connecting them in order forms a rectangle. Find the length of the horizontal side.
   Full step-by-step solution

   Step 1: Plot the points: (-6, 4), (2, 4), (2, -4), (-6, -4). Connecting them in order forms a rectangle. Step 2: Find the length of the horizontal side. The y-coordinates are both 4, so the length is the difference in x-coordinates: 2 - (-6) = 8. Step 3: Find the length of the vertical side. The x-coordinates are both 2, so the length is the difference in y-coordinates: 4 - (-4) = 8. Step 4: Since both sides are 8, the shape is a square. Area of a square = side × side = 8 × 8 = 64. The answer is 64.

6. 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.

7. A rectangle has vertices at (-5, 5), (5, 5), (5, -5), and (-5, -5). What is the area of the rectangle? Answer: 100 Solution: Identify the coordinates of the vertices: A(-5, 5), B(5, 5), C(5, -5), D(-5, -5). Find the width (horizontal distance). Use points A and B: they have the same y-coordinate (5).
   Full step-by-step solution

   Step 1: Identify the coordinates of the vertices: A(-5, 5), B(5, 5), C(5, -5), D(-5, -5). Step 2: Find the width (horizontal distance). Use points A and B: they have the same y-coordinate (5). The distance is |5 - (-5)| = |5 + 5| = 10. Step 3: Find the length (vertical distance). Use points B and C: they have the same x-coordinate (5). The distance is |5 - (-5)| = |5 + 5| = 10. Step 4: Calculate the area: Area = length × width = 10 × 10 = 100. The answer is 100.

8. A triangle has vertices at (-6, 1), (6, 1), and (1, 6). What is the area of the triangle? Answer: 30 Solution: Identify the base. The base can be the side between (-6, 1) and (6, 1). The length of this base is the difference in x-coordinates: 6 - (-6) = 12 units.
   Full step-by-step solution

   Step 1: Identify the base. The base can be the side between (-6, 1) and (6, 1). The length of this base is the difference in x-coordinates: 6 - (-6) = 12 units. Step 2: Find the height. The height is the perpendicular distance from the third vertex (1, 6) to the base line y = 1. The height is 6 - 1 = 5 units. Step 3: Use the area formula for a triangle: Area = 1/2 × base × height = 1/2 × 12 × 5 = 6 × 5 = 30 square units. The answer is 30.

9. A triangle has vertices at (-3, -5), (3, -5), and (1, 3). What is its area in square units? Answer: 24 Solution: Identify the base. The base is between (-3, -5) and (3, -5). Length of base = 3 - (-3) = 6 units.
   Full step-by-step solution

   Step 1: Identify the base. The base is between (-3, -5) and (3, -5). Length of base = 3 - (-3) = 6 units. Step 2: Find the height. The base is at y = -5. The top vertex is at y = 3. Height = 3 - (-5) = 8 units. Step 3: Area of triangle = (1/2) × base × height = (1/2) × 6 × 8 = 24 square units. The answer is 24.