Horizontal and Vertical Distances     CG-L1 Objectives: To solve problems involving the calculation of horizontal and vertical distances. Learning Outcome B-4

Some distances can be difficult to measure physically Some distances can be difficult to measure physically. One way to calculate these distances is to develop a coordinate system and apply coordinate geometry formulas. For most coordinate systems, you arbitrarily assign a convenient fixed reference point, called the Origin (0,0). From that point, you determine coordinates for points of interest, and measure. In 2D situations, you are working with horizontal and vertical distances (not depth as in 3D). Examine the following situation involving one dimension — horizontal distance. Find the distance from the hotel to the ballpark using the following information: each mark is one block. Theory –Horizontal Distance

Therefore, the ballpark is at 0 and the hotel is at 9. If you were to assign numbers to each mark starting at 0, the diagram would be as follows: Therefore, the ballpark is at 0 and the hotel is at 9. You can determine that the distance from the ballpark to the hotel is nine blocks by either counting the marks or subtracting the coordinates of the two landmarks (9 - 0 = 9). You can do the same with vertical distance. The next example adds the dimension of vertical distance. Example 1

Player A broke his finger sliding into second base and had to go to the hospital. How far is it to the hospital from the ballpark? Use the following diagram: Example 2

Coordinate geometry can be used to determine both vertical and horizontal distances. Assume that each number is a street or avenue running east/west and north/south, respectively. You can create the following diagram: Applying your knowledge of points in a coordinate plane, the hospital has coordinates (0,8), the hotel (9,0), and the ballpark (0,0). Understanding that each coordinate is x, paired with y or (x, y), the horizontal distance between two points is the difference of the x-coordinates. In the same way, the vertical distance between two points is the difference of the y-coordinates. Example 3

Find the distance from the ballpark to the hotel. Solution (9,0) (0,0) Hotel - Ball Park 9 - 0 = 9 blocks The distance from the ballpark to the hotel is 9 blocks. Likewise, the vertical distance between ballpark and the hotel would be the difference of the y-coordinates: 0 - 0 = 0 blocks. Example 3

Another landmark has been inserted as the Origin, and the following diagram results: Player A broke his finger sliding into second base and had to go directly to the hospital. How far is it to the hospital from the ballpark? Example 3

The distance from the ballpark to the hotel is 8 blocks. From the original coordinate diagram, the distance would be calculated as follows: (0,8) (0,0) Hospital - Ball Park 8 -0 = 8 blocks The distance from the ballpark to the hotel is 8 blocks. Using the new diagram, the coordinates are: (3, 10) (3, 2) Therefore, the distance from the ballpark to the hospital is 10 – 2 = 8 blocks. What is a general statement (formula) for finding horizontal distance? Vertical Distance? Example 3