Four Quadrants

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Answer Key & Explanations

Four Quadrants · Grade 6 · Worksheet 3

1. (-5, 8) + (3, -12) = ? Answer: (-2, -4) Solution: Add the x-coordinates: -5 + 3 = -2 Add the y-coordinates: 8 + (-12) = 8 - 12 = -4 Combine the results as an ordered pair: (-2, -4) The answer is (-2, -4).
   Full step-by-step solution

   Step 1: Add the x-coordinates: -5 + 3 = -2 Step 2: Add the y-coordinates: 8 + (-12) = 8 - 12 = -4 Step 3: Combine the results as an ordered pair: (-2, -4) The answer is (-2, -4).

2. Liam is designing a treasure map on a coordinate plane where each unit represents 10 meters. He buries a treasure chest at point (-8, 5). He then buries a second treasure chest that is located 12 meters to the left and 7 meters below the first chest. What are the coordinates of the second treasure chest? Answer: (-9.2, 4.3) Solution: Each unit on the coordinate plane represents 10 meters. 1 unit = 10 meters So, 1 meter = 1/10 = 0.1 units. First chest is at (-8, 5) in coordinate units.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the coordinate plane scale** Each unit on the coordinate plane represents 10 meters. That means: 1 unit = 10 meters So, 1 meter = 1/10 = 0.1 units. --- **Step 2: First treasure chest location** First chest is at (-8, 5) in coordinate units. --- **Step 3: Movement for the second chest** The second chest is 12 meters to the left and 7 meters below the first chest. In coordinate units: - 12 meters = 12 × 0.1 = 1.2 units - 7 meters = 7 × 0.1 = 0.7 units --- **Step 4: Apply movement to coordinates** "12 meters to the left" means subtract 1.2 units from the x-coordinate: x_new = -8 - 1.2 = -9.2 "7 meters below" means subtract 0.7 units from the y-coordinate: y_new = 5 - 0.7 = 4.3 --- **Step 5: Final coordinates** Second treasure chest is at (-9.2, 4.3). --- **Answer:** (-9.2, 4.3)

3. 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).

4. Olivia is plotting the locations of buried fossils on a coordinate grid for her science project. She places the first fossil at (-7, 3). The second fossil is located 9 units to the right and 11 units down from the first fossil. What are the coordinates of the second fossil? Answer: (-7 + 9, 3 - 11) = (2, -8) Solution: The first fossil is at (-7, 3). Moving 9 units to the right means adding 9 to the x-coordinate: -7 + 9 = 2. Step 2: Moving 11 units down means subtracting 11 from the y-coordinate: 3 - 11 = -8.
   Full step-by-step solution

   Step 1: The first fossil is at (-7, 3). Moving 9 units to the right means adding 9 to the x-coordinate: -7 + 9 = 2. Step 2: Moving 11 units down means subtracting 11 from the y-coordinate: 3 - 11 = -8. Step 3: The second fossil is at (2, -8). The answer is (2, -8).

5. Sophia is helping her uncle design a small community garden on a coordinate grid. They place the entrance gate at point (-9, 7). The water fountain is located 13 units to the right and 11 units down from the entrance gate. What are the coordinates of the water fountain? Answer: (-9 + 13, 7 - 11) = (4, -4) Solution: The entrance gate is at (-9, 7). Step 2: Moving 13 units to the right means adding 13 to the x-coordinate: -9 + 13 = 4. Step 3: Moving 11 units down means subtracting 11 from the y-coordinate: 7 - 11 = -4.
   Full step-by-step solution

   Step 1: The entrance gate is at (-9, 7). Step 2: Moving 13 units to the right means adding 13 to the x-coordinate: -9 + 13 = 4. Step 3: Moving 11 units down means subtracting 11 from the y-coordinate: 7 - 11 = -4. Step 4: The water fountain is at (4, -4). The answer is (4, -4).

6. (-4, 7) + (3, -9) = ? Answer: (-1, -2) Solution: We are adding two vectors: (-4, 7) and (3, -9). Vector addition is done by adding the corresponding components. Add the x-components.
   Full step-by-step solution

   We are adding two vectors: (-4, 7) and (3, -9). Vector addition is done by adding the corresponding components. Step 1: Add the x-components. The x-component of the first vector is -4. The x-component of the second vector is 3. So, x-component of the sum = (-4) + 3 = -1. Step 2: Add the y-components. The y-component of the first vector is 7. The y-component of the second vector is -9. So, y-component of the sum = 7 + (-9) = 7 - 9 = -2. Step 3: Combine the results. The resulting vector is (-1, -2). Final answer: (-1, -2)

7. Charlotte is helping her family plan a new garden layout on a coordinate grid. They decide to place a birdbath at point (-7, 2). The rose bush is located 12 units directly to the right and 9 units directly down from the birdbath. What are the coordinates of the rose bush? Answer: (5, -7) Solution: Start with the birdbath's coordinates: (-7, 2). Moving 12 units to the right means adding 12 to the x-coordinate: -7 + 12 = 5. Moving 9 units down means subtracting 9 from the y-coordinate: 2 - 9 = -7.
   Full step-by-step solution

   Step 1: Start with the birdbath's coordinates: (-7, 2). Step 2: Moving 12 units to the right means adding 12 to the x-coordinate: -7 + 12 = 5. Step 3: Moving 9 units down means subtracting 9 from the y-coordinate: 2 - 9 = -7. Step 4: The rose bush is located at (5, -7). The answer is (5, -7).

8. (-3, 4) + (7, -9) = ? Answer: (4, -5) Solution: We are adding two vectors: (-3, 4) and (7, -9). 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 (7, -9). 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 = 7 and y-component = -9. Step 2: Add the x-components together. x_total = (-3) + 7 x_total = 4 Step 3: Add the y-components together. y_total = 4 + (-9) y_total = 4 - 9 y_total = -5 Step 4: Combine the results into a single vector. Result = (x_total, y_total) = (4, -5) Final answer: (4, -5)