Four Quadrants

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Answer Key & Explanations

Four Quadrants · Grade 6 · Worksheet 1

1. Liam is designing a treasure map on a coordinate plane where each unit represents 10 meters. He buries a treasure at point (-3, 4). Then, he buries a second treasure 50 meters directly east and 30 meters directly south of the first treasure. What are the coordinates of the second treasure? Answer: (2, 1) Solution: Each unit on the coordinate plane represents 10 meters. So, if something moves 1 unit east, that's 10 meters east. If something moves 1 unit south, that's 10 meters south.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the coordinate system** Each unit on the coordinate plane represents 10 meters. So, if something moves 1 unit east, that's 10 meters east. If something moves 1 unit south, that's 10 meters south. --- **Step 2: First treasure location** First treasure is at (-3, 4). --- **Step 3: Movement for second treasure** - 50 meters east: Since 1 unit = 10 meters, 50 meters = 50/10 = 5 units. Moving east means increasing the x-coordinate. So: new x = -3 + 5 = 2. - 30 meters south: 30 meters = 30/10 = 3 units. Moving south means decreasing the y-coordinate. So: new y = 4 - 3 = 1. --- **Step 4: Coordinates of second treasure** (2, 1) --- **Step 5: Check** From (-3, 4) to (2, 1): Change in x: 5 units → 50 meters east ✓ Change in y: -3 units → 30 meters south ✓ --- **Final answer:** (2, 1)

2. Ava is using a coordinate plane to design a logo for her school's robotics club. She places the center of a gear at point (-6, 1). She then places a bolt 6 units directly to the right and 7 units directly down from the gear. What are the coordinates of the bolt? Answer: (0, -6) Solution: The gear's center is at (-6, 1). Moving 6 units to the right means adding 6 to the x-coordinate: -6 + 6 = 0. Moving 7 units down means subtracting 7 from the y-coordinate: 1 - 7 = -6.
   Full step-by-step solution

   Step 1: The gear's center is at (-6, 1). Step 2: Moving 6 units to the right means adding 6 to the x-coordinate: -6 + 6 = 0. Step 3: Moving 7 units down means subtracting 7 from the y-coordinate: 1 - 7 = -6. Step 4: The bolt is at (0, -6). The answer is (0, -6).

3. Isabella is plotting the locations of four different landmarks in her town on a coordinate plane. The town hall is at point (-7, 5). The library is located 12 units to the right and 8 units down from the town hall. The park is at point (2, -7). The school is at point (-2, 3). Which landmark is in Quadrant II, and what are the coordinates of the library? Answer: Town hall is in Quadrant II; library is at (5, -3) Solution: Identify which landmarks are in Quadrant II. Quadrant II has points where x is negative and y is positive. Town hall: (-7, 5) — x is negative, y is positive, so it is in Quadrant II.
   Full step-by-step solution

   Step 1: Identify which landmarks are in Quadrant II. Quadrant II has points where x is negative and y is positive. Town hall: (-7, 5) — x is negative, y is positive, so it is in Quadrant II. Library: We will find its coordinates in the next step. Park: (2, -7) — x is positive, y is negative, so it is in Quadrant IV. School: (-2, 3) — x is negative, y is positive, so it is also in Quadrant II. Both town hall and school are in Quadrant II, but the question asks for which landmark is in Quadrant II, and the town hall is one of them (the school is also there, but the primary answer is town hall since it was listed first). Step 2: Find the coordinates of the library. Town hall is at (-7, 5). Moving 12 units to the right: add 12 to the x-coordinate: -7 + 12 = 5. Moving 8 units down: subtract 8 from the y-coordinate: 5 - 8 = -3. So the library is at (5, -3). The answer is: Town hall is in Quadrant II; library is at (5, -3).

4. 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.

5. Ava is helping her uncle map out a new garden layout on a coordinate grid. They place the birdbath at point (-8, 9) and the rose bush at point (7, -5). Ava says the midpoint between the birdbath and the rose bush would be the perfect spot for a small fountain. What are the coordinates of the fountain? Answer: (-0.5, 2) Solution: Identify the coordinates: birdbath at (-8, 9) and rose bush at (7, -5). Find the average of the x-coordinates: (-8 + 7) / 2 = (-1) / 2 = -0.5. Find the average of the y-coordinates: (9 + (-5)) / 2 = (4) / 2 = 2.
   Full step-by-step solution

   Step 1: Identify the coordinates: birdbath at (-8, 9) and rose bush at (7, -5). Step 2: Find the average of the x-coordinates: (-8 + 7) / 2 = (-1) / 2 = -0.5. Step 3: Find the average of the y-coordinates: (9 + (-5)) / 2 = (4) / 2 = 2. Step 4: The midpoint coordinates are (-0.5, 2). The fountain should be placed at (-0.5, 2).

6. Mere is designing a garden layout on a coordinate plane where each unit represents 1 meter. She places a fountain at point (-8, 6). A birdbath is located 14 meters to the left and 10 meters down from the fountain. What are the coordinates of the birdbath? Answer: (-22, -4) Solution: The fountain is at (-8, 6). Moving 14 meters to the left means subtracting 14 from the x-coordinate: -8 - 14 = -22. Step 2: Moving 10 meters down means subtracting 10 from the y-coordinate: 6 - 10 = -4.
   Full step-by-step solution

   Step 1: The fountain is at (-8, 6). Moving 14 meters to the left means subtracting 14 from the x-coordinate: -8 - 14 = -22. Step 2: Moving 10 meters down means subtracting 10 from the y-coordinate: 6 - 10 = -4. Step 3: The birdbath is at (-22, -4). The answer is (-22, -4).

7. A treasure map shows a rectangular treasure chest located on a coordinate plane. The chest's corners are at points (-5, 2), (3, 2), (3, -4), and (-5, -4). If each unit on the grid represents 2 feet, what is the area of the treasure chest in square feet? Answer: 192 Solution: Identify the coordinates of the rectangle's corners A = (-5, 2) B = (3, 2) C = (3, -4) D = (-5, -4) Points A and B are both at y = 2, so AB is horizontal.
   Full step-by-step solution

   Let's solve this step-by-step. --- **Step 1: Identify the coordinates of the rectangle's corners** The corners are: A = (-5, 2) B = (3, 2) C = (3, -4) D = (-5, -4) --- **Step 2: Find the length and width in coordinate units** Points A and B are both at y = 2, so AB is horizontal. A = (-5, 2), B = (3, 2) Length in units = 3 - (-5) = 3 + 5 = 8 units. Points B and C are both at x = 3, so BC is vertical. B = (3, 2), C = (3, -4) Width in units = 2 - (-4) = 2 + 4 = 6 units. So the rectangle is 8 units long and 6 units wide in the coordinate plane. --- **Step 3: Convert units to feet** Each unit on the grid = 2 feet. So: Length in feet = 8 units × 2 feet/unit = 16 feet Width in feet = 6 units × 2 feet/unit = 12 feet --- **Step 4: Calculate area in square feet** Area = length × width Area = 16 × 12 = 192 square feet --- **Final Answer:** 192