Assignment 10
Two distinct nonvertical lines are parallel if and only if they have the same slope. Logically, we could write
Note that distinct lines that have an undefined slope are also parallel. (This should make sense!) Now, what about perpendicular lines? We know a horizontal line is perpendicular to a vertical line. So, we can conclude that if line L_{1} has an undefined slope and line L_{2} has a slope = 0, then L_{1} is perpendicular to L_{2}. What if lines do not have an undefined slope? Here is an important mathematical reality:
You can use geometry to demonstrate that this result is reasonable. ================================ Important points made in the reading for Section 2.4.
Math is a language. Math is power. For over 2000 years, humans have developed and used mathematics in attempts to understand the universe in which they existed. Our universe has a mathematical design.
Problem: Write the equation of the line that passes through (1,5) and is perpendicular to y = -(1/2)x + 18. Solution (with communication): The slope of the desired line is 2. The equation of the line is y - 5 = 2(x - 1) ==> y = 2x + 3. Problem: What is the equation of the line containing (4,7) that is perpendicular to the line passing through (-3,9) and (21,9). Solution (with communication): The line containing (-3,9) and (21,9) is horizontal. The equation of the line is y = 9 and the slope is 0. A line perpendicular to this line is vertical, and has no slope. The equation of the vertical line containing (4,7) is x = 4. Problem: x varies directly with y, and x = 32 when y = 2. What is the value of y when x = 128? Solution (with communication): Premise ==> x = Ky for some constant K. x = 32 when y = 2 ==> 32 = K(2) ==> K = 16. Hence x = 16y. x = 128 ==> 128 = 16(y) ==> y = 128/16 = 8. Problem: Tell whether the data show direct variation.
Solution (with communication): y = Kx ==> y/x = K for direct variation. In this case, we have 8/2 = 36/9 = 40/10 = 44/11 = 4, but 60/12 = 5. Hence the data set does not show direct variation. [Note: The first four points are contained on the line y = 4x, but (12,60) is not on this line.] |