Four Quadrants

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Answer Key & Explanations

Four Quadrants · Grade 6 · Worksheet 3

1. Emma is plotting the locations of landmarks for a city planning project on a coordinate plane. She places the town hall at point (-5, 7). The library is located 9 units to the right and 11 units down from the town hall. The park is located 6 units to the left and 4 units up from the library. What are the coordinates of the park? Answer: (-2, 0) Solution: Start with the town hall at (-5, 7). Find the library. Move 9 units right: -5 + 9 = 4.
   Full step-by-step solution

   Step 1: Start with the town hall at (-5, 7). Step 2: Find the library. Move 9 units right: -5 + 9 = 4. Move 11 units down: 7 - 11 = -4. So the library is at (4, -4). Step 3: Find the park from the library. Move 6 units left: 4 - 6 = -2. Move 4 units up: -4 + 4 = 0. So the park is at (-2, 0). The answer is (-2, 0).

2. (-2, 5) + (3, -7) = ? Answer: (1, -2) Solution: Identify the x-coordinates: -2 and 3. Add them: -2 + 3 = 1. Identify the y-coordinates: 5 and -7.
   Full step-by-step solution

   Step 1: Identify the x-coordinates: -2 and 3. Add them: -2 + 3 = 1. Step 2: Identify the y-coordinates: 5 and -7. Add them: 5 + (-7) = -2. Step 3: Combine the results to form the new coordinate: (1, -2). The answer is (1, -2).

3. Emma is helping her aunt design a garden layout on a coordinate grid. They place a fountain at point (-5, 10). A flower bed is located 15 units to the right and 20 units down from the fountain. A bench is placed exactly halfway between the fountain and the flower bed. What are the coordinates of the bench? Answer: (2.5, 0) Solution: Find the coordinates of the flower bed. The fountain is at (-5, 10). Moving 15 units right means adding 15 to the x-coordinate: -5 + 15 = 10.
   Full step-by-step solution

   Step 1: Find the coordinates of the flower bed. The fountain is at (-5, 10). Moving 15 units right means adding 15 to the x-coordinate: -5 + 15 = 10. Moving 20 units down means subtracting 20 from the y-coordinate: 10 - 20 = -10. So the flower bed is at (10, -10). Step 2: Find the midpoint between the fountain and the flower bed. The midpoint formula is: ((x1 + x2)/2, (y1 + y2)/2) For x: (-5 + 10)/2 = 5/2 = 2.5 For y: (10 + (-10))/2 = 0/2 = 0 Step 3: The bench is at (2.5, 0). The answer is (2.5, 0).

4. (-3, 4) + (5, -2) = ? Answer: (2, 2) Solution: We are adding two vectors: (-3, 4) and (5, -2). Identify the components of each vector. The first vector has x-component = -3 and y-component = 4.
   Full step-by-step solution

   We are adding two vectors: (-3, 4) and (5, -2). Step 1: Identify the components of each vector. The first vector has x-component = -3 and y-component = 4. The second vector has x-component = 5 and y-component = -2. Step 2: Add the x-components together. x_total = (-3) + (5) = 2 Step 3: Add the y-components together. y_total = (4) + (-2) = 4 - 2 = 2 Step 4: Combine the results into a single vector. The resulting vector is (x_total, y_total) = (2, 2) Therefore, (-3, 4) + (5, -2) = (2, 2).

5. A rectangle is drawn on a coordinate plane with vertices at (-5, 2), (3, 2), (3, -4), and (-5, -4). What is the area of this rectangle? Answer: 48 Solution: Identify the vertices and their coordinates. A = (-5, 2) B = (3, 2) C = (3, -4) D = (-5, -4) Understand the rectangle's sides. Points A and B have the same y-coordinate (y = 2), so AB is a horizontal segment.
   Full step-by-step solution

   Step 1: Identify the vertices and their coordinates. The vertices are: A = (-5, 2) B = (3, 2) C = (3, -4) D = (-5, -4) Step 2: Understand the rectangle's sides. Points A and B have the same y-coordinate (y = 2), so AB is a horizontal segment. Points B and C have the same x-coordinate (x = 3), so BC is a vertical segment. Thus, AB is the length and BC is the width (or vice versa). Step 3: Find the length of side AB. A = (-5, 2), B = (3, 2) Length = difference in x-coordinates = 3 - (-5) = 3 + 5 = 8. Step 4: Find the length of side BC. B = (3, 2), C = (3, -4) Width = difference in y-coordinates = 2 - (-4) = 2 + 4 = 6. Step 5: Calculate the area of the rectangle. Area = length × width = 8 × 6 = 48. Step 6: Conclusion. The area of the rectangle is 48 square units.

6. Emma is designing a video game map on a coordinate plane. She places a treasure chest at point (-7, 3). She then places a magic portal that is 9 units to the left and 5 units down from the treasure chest. What are the coordinates of the magic portal? Answer: (-16, -2) Solution: The treasure chest is at (-7, 3). Step 2: Moving 9 units to the left means subtracting 9 from the x-coordinate: -7 - 9 = -16. Step 3: Moving 5 units down means subtracting 5 from the y-coordinate: 3 - 5 = -2.
   Full step-by-step solution

   Step 1: The treasure chest is at (-7, 3). Step 2: Moving 9 units to the left means subtracting 9 from the x-coordinate: -7 - 9 = -16. Step 3: Moving 5 units down means subtracting 5 from the y-coordinate: 3 - 5 = -2. Step 4: The magic portal is at (-16, -2). The answer is (-16, -2).

7. Ava is plotting a hidden treasure on a coordinate plane. The treasure is located at the point that is the reflection of point A across the x-axis, then the reflection of that result across the y-axis. Point A is at (-11, 6). What are the coordinates of the treasure? Answer: (11, -6) Solution: Start with point A at (-11, 6). Reflect across the x-axis. The x-coordinate stays the same (-11), and the y-coordinate changes sign (6 becomes -6).
   Full step-by-step solution

   Step 1: Start with point A at (-11, 6). Step 2: Reflect across the x-axis. The x-coordinate stays the same (-11), and the y-coordinate changes sign (6 becomes -6). So the point after this reflection is (-11, -6). Step 3: Now reflect (-11, -6) across the y-axis. The y-coordinate stays the same (-6), and the x-coordinate changes sign (-11 becomes 11). So the final point is (11, -6). Step 4: The treasure is at (11, -6).

8. Liam is designing a treasure map on a coordinate plane. He places a treasure chest at point (-4, 5) and a clue at point (3, -2). If Liam needs to place a key exactly halfway between the treasure chest and the clue, what are the coordinates of the key? Answer: (-0.5, 1.5) Solution: To find the point exactly halfway between two given points, we calculate the midpoint. The midpoint formula is: Midpoint = ( (x1 + x2)/2 , (y1 + y2)/2 ) Identify the coordinates of the two points.
   Full step-by-step solution

   To find the point exactly halfway between two given points, we calculate the midpoint. The midpoint formula is: Midpoint = ( (x1 + x2)/2 , (y1 + y2)/2 ) Step 1: Identify the coordinates of the two points. - Treasure chest: (x1, y1) = (-4, 5) - Clue: (x2, y2) = (3, -2) Step 2: Calculate the x-coordinate of the midpoint. - Add the x-coordinates: x1 + x2 = -4 + 3 = -1 - Divide the sum by 2: -1 / 2 = -1/2 = -0.5 Step 3: Calculate the y-coordinate of the midpoint. - Add the y-coordinates: y1 + y2 = 5 + (-2) = 5 - 2 = 3 - Divide the sum by 2: 3 / 2 = 3/2 = 1.5 Step 4: Combine the results. The coordinates of the key (the midpoint) are (-0.5, 1.5). This means the key is located at x = -0.5 and y = 1.5 on the coordinate plane, exactly halfway between the treasure chest and the clue.