Coordinate Polygons

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Answer Key & Explanations

Coordinate Polygons · Grade 6 · Worksheet 3

1. A polygon has vertices at (-5, 0), (0, 5), (5, 0), (0, -5). What is the area of this polygon? Answer: 50 Solution: Plot the vertices: (-5, 0), (0, 5), (5, 0), (0, -5). The shape is a rhombus (a square rotated 45 degrees). Find the lengths of the diagonals.
   Full step-by-step solution

   Step 1: Plot the vertices: (-5, 0), (0, 5), (5, 0), (0, -5). The shape is a rhombus (a square rotated 45 degrees). Step 2: Find the lengths of the diagonals. Diagonal 1 connects (-5, 0) and (5, 0): length = 5 - (-5) = 10 units. Diagonal 2 connects (0, 5) and (0, -5): length = 5 - (-5) = 10 units. Step 3: Area of a rhombus = (d1 × d2) / 2 = (10 × 10) / 2 = 100 / 2 = 50 square units. The answer is 50.

2. A quadrilateral has vertices at (-2, 4), (6, 4), (6, -3), and (-2, -3) on a coordinate plane. What is the perimeter of this quadrilateral in units? Answer: 30 Solution: Plot the points mentally: (-2,4), (6,4), (6,-3), (-2,-3) Notice that the x-coordinates of the first two points are different but y-coordinates are the same (both 4), so this is a horizontal line segment Calculate the length of the top side: 6 - (-2) = 6 + 2 = 8 units Notice that the…
   Full step-by-step solution

   Step 1: Plot the points mentally: (-2,4), (6,4), (6,-3), (-2,-3) Step 2: Notice that the x-coordinates of the first two points are different but y-coordinates are the same (both 4), so this is a horizontal line segment Step 3: Calculate the length of the top side: 6 - (-2) = 6 + 2 = 8 units Step 4: Notice that the y-coordinates of the second and third points are different but x-coordinates are the same (both 6), so this is a vertical line segment Step 5: Calculate the length of the right side: 4 - (-3) = 4 + 3 = 7 units Step 6: Since opposite sides of a rectangle are equal, the bottom side is also 8 units and the left side is also 7 units Step 7: Calculate the perimeter: 8 + 7 + 8 + 7 = 30 units The answer is 30.

3. 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.

4. Liam plots a quadrilateral on a coordinate plane with vertices at A(-15, 10), B(25, 10), C(25, -20), and D(-15, -20). 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: 600 Solution: Plot the points: A(-15, 10), B(25, 10), C(25, -20), D(-15, -20). The x-coordinates are -15 and 25, and the y-coordinates are 10 and -20, so the quadrilateral is a rectangle.
   Full step-by-step solution

   Step 1: Plot the points: A(-15, 10), B(25, 10), C(25, -20), D(-15, -20). The x-coordinates are -15 and 25, and the y-coordinates are 10 and -20, so the quadrilateral is a rectangle. Step 2: Triangle ABC has vertices A(-15, 10), B(25, 10), and C(25, -20). Step 3: Side AB is horizontal from x = -15 to x = 25. Length AB = 25 - (-15) = 25 + 15 = 40 units. Step 4: The height of triangle ABC is the vertical distance from point C to line AB. Point C has y = -20, and line AB has y = 10. Height = 10 - (-20) = 10 + 20 = 30 units. Step 5: Area of triangle = 1/2 × base × height = 1/2 × 40 × 30 = 20 × 30 = 600 square units. The answer is 600.

5. 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.

6. (-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.

7. Mere plots a triangle on a coordinate plane with vertices at A(-25, 18), B(25, 18), and C(0, -18). What is the area of this triangle? Answer: 900 Solution: Identify the base. Points A(-25, 18) and B(25, 18) have the same y-coordinate, so side AB is horizontal. Length of AB = 25 - (-25) = 25 + 25 = 50 units.
   Full step-by-step solution

   Step 1: Identify the base. Points A(-25, 18) and B(25, 18) have the same y-coordinate, so side AB is horizontal. Length of AB = 25 - (-25) = 25 + 25 = 50 units. Step 2: Find the height. The height is the vertical distance from point C(0, -18) to the line AB (y = 18). Height = 18 - (-18) = 18 + 18 = 36 units. Step 3: Calculate the area of the triangle. Area = 1/2 × base × height = 1/2 × 50 × 36 = 25 × 36 = 900 square units. The answer is 900.

8. Sophia plots a quadrilateral on a coordinate plane with vertices at A(-18, 20), B(18, 20), C(18, -14), and D(-18, -14). She then draws a diagonal from point A to point C, dividing the quadrilateral into two triangles. What is the area of triangle ABC? Answer: 612 Solution: Plot the points: A(-18, 20), B(18, 20), C(18, -14), D(-18, -14). The x-coordinates are -18 and 18, and the y-coordinates are 20 and -14, so the quadrilateral is a rectangle.
   Full step-by-step solution

   Step 1: Plot the points: A(-18, 20), B(18, 20), C(18, -14), D(-18, -14). The x-coordinates are -18 and 18, and the y-coordinates are 20 and -14, so the quadrilateral is a rectangle. Step 2: Triangle ABC has vertices A(-18, 20), B(18, 20), and C(18, -14). Step 3: Side AB is horizontal from x = -18 to x = 18. Length of AB = 18 - (-18) = 18 + 18 = 36 units. Step 4: The height of triangle ABC is the vertical distance from point C to line AB. Point C has y = -14, and line AB has y = 20. Height = 20 - (-14) = 20 + 14 = 34 units. Step 5: Area of triangle = 1/2 × base × height = 1/2 × 36 × 34 = 18 × 34 = 612 square units. The answer is 612.

9. Mere draws a quadrilateral on a coordinate plane with vertices at A(-18, 14), B(22, 14), C(22, -16), and D(-18, -16). 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: 600 Solution: Identify the vertices of triangle ABC: A(-18, 14), B(22, 14), and C(22, -16). Notice that A and B have the same y-coordinate (14), so side AB is horizontal.
   Full step-by-step solution

   Step 1: Identify the vertices of triangle ABC: A(-18, 14), B(22, 14), and C(22, -16). Step 2: Notice that A and B have the same y-coordinate (14), so side AB is horizontal. Find the length of AB: from x = -18 to x = 22, so AB = 22 - (-18) = 22 + 18 = 40 units. Step 3: The height of triangle ABC is the vertical distance from point C to the line through A and B (y = 14). Point C has y = -16, so the height = 14 - (-16) = 14 + 16 = 30 units. Step 4: Area of triangle = 1/2 × base × height = 1/2 × 40 × 30 = 20 × 30 = 600 square units. The answer is 600.