Applications with the Coordinate Plane Lesson 4.04

After completing this lesson, you will be able to say: I can plot pairs of values that represent equivalent ratios on the coordinate plane. I can solve real-world and mathematical problems involving ratio and rate using coordinates

Using the Coordinate Plane to solve a real world problem We can use the coordinate plane to help solve real world problems Let’s look at some examples

Example 1 Cooper's parents just bought a new swing set for the backyard. His dad decided to put four garden wind spinners around it. He drew a coordinate plane to help him see where to place the spinners. Help Cooper's dad to place the spinners at the following points: (−7, 5), (7, 5), (7, −3), and (−7, −3).

Example 2 The local park is creating a new walking trail. The town mayor asks the town architect to design the trail for the next meeting. The trail needs to start and end near the park entrance. The architect plots the points on the coordinate grid to offer his proposal to the mayor.

Example 2 plotting the points

Finding Measurements Quadrant 1 We can use the coordinate plane to find the measurement of line segments. Method 1: Counting spaces By counting the spaces between the points, you can determine how far two points are from each other. You will see that the door is 4 units. Start on the ordered pair (6, 15), and count the number of spaces (or units) from that point to (10, 15).

Finding Measurements – Quadrant 1 Method 2 - SubtractionWith the subtraction method, you just need the coordinates that are different. Identify the first and last coordinates of the line segment. First point: (6, 15) Second point: (10, 15) Notice that the x-coordinates are different and the y-coordinates are the same. Take the values of the x-coordinates and subtract them (remember to put the number with the larger value first). 10 − 6 = 4 units This shows that the door is 4 units long on the grid.

Try it Use the coordinate plane to find the measurements of the cabinet

Check your work Method 1: Counting Spaces From (11, 11) to (14, 11), there are 3 units. From (14, 11) to (14, 2), there are 9 units. Because it is a rectangle, you know the other two sides will be the same length. Method 2: Subtracting Remember, subtract the coordinates that are different. From (11, 11) to (14, 11): 14 − 11 = 3 units. From (14, 11) to (14, 2): 11 − 2 = 9 units. Because it is a rectangle, you know the other two sides will be the same length.

Finding Measurements – Quadrant 2 Using the coordinate plane, find the measurements of the TV Method 1: Counting Spaces From (−13, 12) to (−12, 12), there is 1 unit. From (−12, 12) to (−12, 3), there are 9 units. Because it is a rectangle, you know the other two sides will be the same length.

Finding Measurement – Quadrant 2 Method 2: Subtracting In the second quadrant, some of the values are negative. So how would you subtract? Since you are finding the distance, remember distance is always positive. This means you have to take the absolute value of each negative value before subtracting. Review absolute value facts if you need to. From (−13, 12) to (−12, 12), |−13| − |−12| = 13 − 12 = 1 unit From (−12, 12) to (−12, 3), 12 − 3 = 9 units. Because it is a rectangle, you know the other two sides will be the same length

Try it! Use the coordinate plane to find the measurements of the video game

Check your work The video game area is a square, so all sides are the same length. Just work with any two corners. From (−13, −8) to (−9, −8), there are 4 units.

Try it Use the coordinate plane to find the measurements of the air hockey table

Check your work Air Hockey Table From (5, −4) to (10, −4): 10 − 5 = 5 units. From (10, −4) to (10, −13) = |−13| − |−4| = 13 − 4 = 9 units.

Finding Measurements: Multiple Quadrants What about the foosball table? Notice it's in more than one quadrant.

Finding Measurements: Multiple Quadrants Method 1: counting spaces You can still count the spaces to find the lengths. From (−4, −2) to (2, −2), there are 6 units. From (2, −2) to (2, −4), there are 2 units.

Try it Let's calculate the distance between each of the points and then add them to determine the entire length of the trail.

Check your work Using the counting method after all points are plotted. From (−5, 7) to (−5, 3), there are 4 units. From (−5, 3) to (5, 3), there are 10 units. From (5, 3) to (5, −5), there are 8 units. From (5, −5) to (−7, −5), there are 12 units. From (−7, −5) to (−7, 7), there are 12 units = 46 total units

Finding Measurements: Multiple Quadrants Method 2: Subtraction When a line segment crosses over an quadrant, you have to find the distance the endpoints are from the axis that is crossed. Since the axes have a value of 0, find the absolute value of the coordinates that are different in the two ordered pairs. Finally, instead of subtracting, you add the two distances. From (−4, −2) to (2, −2), notice this goes from Quadrant 3 to Quadrant 4. This goes across the axis, so you add the absolute value of the coordinates that are different. Distance would be |−4| + |2| = = 6 units. The distance from (−4, −4) to (−4, −2) is different. No axis is crossed, so you subtract. |−4| − |−2| = 4 − 2 = 2 units.

Now that you completed this lesson, you should be able to say: I can solve real-world problems by graphing points in all four quadrants of a coordinate plane. I can use coordinates to determine distances between points.