Question

problem of the day 9.2.4
topic: parallel and perpendicular lines
understanding how parallel and perpendicular lines behave is an important part of coordinate
geometry. in the coordinate plane,
parallel lines - have the same slope but different y - intercepts
perpendicular lines - have slopes that are negative reciprocals of each other
these concepts are essential for solving equations of lines, analyzing angles, and proving
geometric relationships.
part 1. fill in the blank
fill in the blank with the correct words in the word bank.

1. lines are ________ if they have the same slope but different y - intercepts.
2. two lines are ________ if the product of their slopes is - 1.
3. the slope of a line perpendicular to a line with slope 2/3 is ________
4. the equation y = 4x + 3 is parallel to the line y = ________
5. the equation y = - 1/2x + 3 is perpendicular to the line y = ________

word bank
negative reciprocal perpendicular 2x - 5 additive inverse
parallel - 4x + 7 - 3/2 - 1/2x + 8

Explanation:

Step1: Analyze the first blank

According to the definition of parallel lines (same slope, different y - intercept), when lines have the same slope but different y - intercepts, they are parallel.

Step2: Analyze the second blank

For two lines to be perpendicular, the product of their slopes is - 1. So, if the product of slopes is - 1, the lines are perpendicular.

Step3: Calculate the slope for the third blank

If a line has slope \(m = \frac{2}{3}\), the slope of a perpendicular line is the negative reciprocal. The negative reciprocal of \(\frac{2}{3}\) is \(-\frac{3}{2}\) (using the formula \(m_{perp}=-\frac{1}{m}\) where \(m\) is the slope of the original line).

Step4: Determine the equation for the fourth blank

The line \(y = 4x+3\) has a slope of 4. A parallel line must have the same slope. Among the options, \(y = 4x + 7\) has a slope of 4 (in the form \(y=mx + b\), \(m\) is the slope).

Step5: Find the equation for the fifth blank

The line \(y=-\frac{1}{2}x + 3\) has a slope of \(-\frac{1}{2}\). The slope of a perpendicular line is the negative reciprocal. The negative reciprocal of \(-\frac{1}{2}\) is 2. The line \(y = 2x-5\) has a slope of 2 (in the form \(y = mx + b\), \(m\) is the slope).

Answer:

1. Parallel
2. Perpendicular
3. -3/2
4. 4x + 7
5. 2x - 5