Coordinate Polygons

lessonbunny.com

Answer Key & Explanations

Coordinate Polygons · Grade 6 · Worksheet 1

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. 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.

3. Sophia plots a polygon on a coordinate plane with vertices at A(-16, 11), B(26, 11), C(26, -21), and D(-16, -21). She then draws a line segment from point A to point C, dividing the polygon into two triangles. What is the area of triangle ABC? Answer: 672 Solution: Identify the shape. The vertices A(-16, 11), B(26, 11), C(26, -21), and D(-16, -21) form a rectangle because opposite sides have equal lengths and are parallel. Focus on triangle ABC.
   Full step-by-step solution

   Step 1: Identify the shape. The vertices A(-16, 11), B(26, 11), C(26, -21), and D(-16, -21) form a rectangle because opposite sides have equal lengths and are parallel. Step 2: Focus on triangle ABC. Its vertices are A(-16, 11), B(26, 11), and C(26, -21). Step 3: Use side AB as the base. Points A and B have the same y-coordinate (11), so AB is horizontal. Length of AB = 26 - (-16) = 26 + 16 = 42 units. Step 4: Find the height of triangle ABC. The height is the vertical distance from point C to line AB (y = 11). Point C has y = -21. Height = 11 - (-21) = 11 + 21 = 32 units. Step 5: Calculate the area. Area = 1/2 × base × height = 1/2 × 42 × 32 = 21 × 32 = 672 square units. The answer is 672.

4. Kaia plots a quadrilateral on a coordinate plane with vertices at A(-14, 10), B(10, 10), C(10, -16), and D(-14, -16). She then draws a diagonal from point B to point D, dividing the quadrilateral into two triangles. What is the area of triangle BCD? Answer: 312 Solution: Identify the vertices of triangle BCD: B(10, 10), C(10, -16), D(-14, -16). Notice that points B and C share the same x-coordinate (10), so side BC is vertical. Length of BC = 10 - (-16) = 10 + 16 = 26 units.
   Full step-by-step solution

   Step 1: Identify the vertices of triangle BCD: B(10, 10), C(10, -16), D(-14, -16). Step 2: Notice that points B and C share the same x-coordinate (10), so side BC is vertical. Length of BC = 10 - (-16) = 10 + 16 = 26 units. Step 3: Points C and D share the same y-coordinate (-16), so side CD is horizontal. Length of CD = 10 - (-14) = 10 + 14 = 24 units. Step 4: Triangle BCD is a right triangle with legs BC and CD meeting at point C. The area of a right triangle is 1/2 × leg1 × leg2. Step 5: Area = 1/2 × 26 × 24 = 13 × 24 = 312 square units. The answer is 312.

5. 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.

6. (-15 + 8) × 3 = ? Answer: -21 Solution: First, solve the operation inside the parentheses. We have (-15 + 8). Adding a positive number to a negative number means we subtract their absolute values and keep the sign of the larger number.
   Full step-by-step solution

   Step 1: First, solve the operation inside the parentheses. We have (-15 + 8). Adding a positive number to a negative number means we subtract their absolute values and keep the sign of the larger number. 15 - 8 = 7. Since 15 is larger and negative, the result is negative. So, (-15 + 8) = -7. Step 2: Now we have (-7) × 3. Multiplying a negative number by a positive number gives a negative result. 7 × 3 = 21, so (-7) × 3 = -21. Final Answer: -21

7. Emma plots a quadrilateral on a coordinate plane with vertices at A(-15, 9), B(15, 9), C(15, -13), and D(-15, -13). She 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: 330 Solution: Plot the points: A(-15, 9), B(15, 9), C(15, -13), D(-15, -13). The x-coordinates are -15 and 15, and the y-coordinates are 9 and -13, so the quadrilateral is a rectangle.
   Full step-by-step solution

   Step 1: Plot the points: A(-15, 9), B(15, 9), C(15, -13), D(-15, -13). The x-coordinates are -15 and 15, and the y-coordinates are 9 and -13, so the quadrilateral is a rectangle. Step 2: Triangle ABC has vertices A(-15, 9), B(15, 9), and C(15, -13). Step 3: Side AB is horizontal from x = -15 to x = 15. Length AB = 15 - (-15) = 15 + 15 = 30 units. Step 4: The height of triangle ABC is the vertical distance from point C to line AB. Point C has y = -13, and line AB has y = 9. Height = 9 - (-13) = 9 + 13 = 22 units. Step 5: Area of triangle = 1/2 × base × height = 1/2 × 30 × 22 = 15 × 22 = 330 square units. The answer is 330.

8. A rectangular garden is plotted on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). A triangular flower bed is created inside the garden by connecting points (2, 1), (8, 1), and (5, 5). What percentage of the garden's total area is occupied by the triangular flower bed? Answer: 50% Solution: (2, 1), (8, 1), (8, 5), (2, 5). - (2, 1) to (8, 1) is bottom side (length 6). - (8, 1) to (8, 5) is right side (height 4).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the shapes.** The rectangular garden has vertices: (2, 1), (8, 1), (8, 5), (2, 5). Plotting these: - (2, 1) to (8, 1) is bottom side (length 6). - (8, 1) to (8, 5) is right side (height 4). - (8, 5) to (2, 5) is top side (length 6). - (2, 5) to (2, 1) is left side (height 4). So the rectangle's width = 8 - 2 = 6, height = 5 - 1 = 4. --- **Step 2: Area of the rectangle.** Area_rect = width × height = 6 × 4 = 24. --- **Step 3: Identify the triangle's vertices.** Triangle: (2, 1), (8, 1), (5, 5). Base: from (2, 1) to (8, 1) is length 6. Height: from base y = 1 to top vertex (5, 5) is vertical distance = 5 - 1 = 4. --- **Step 4: Area of the triangle.** Area_triangle = (1/2) × base × height = (1/2) × 6 × 4 = (1/2) × 24 = 12. --- **Step 5: Percentage of garden occupied by triangle.** Percentage = (Area_triangle / Area_rect) × 100 = (12 / 24) × 100 = (1/2) × 100 = 50%. --- **Step 6: Conclusion.** The triangular flower bed occupies 50% of the garden's area. --- **Final answer:** 50%