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# High School: Geometry

### Expressing Geometric Properties with Equations HSG-GPE.B.5

5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

If you live within driving distance of one of the top five ski resorts in the lower forty-eight, hitting the slopes can be a year round pastime. If you don't, it's still possible to ski year round,

Regardless of their skills on the slopes, we hope students won't experience epic fails when it comes to the slopes. Well, geometric slopes, anyway.

Students are expected to know that lines in the same plane can fall into one of three categories: parallel, perpendicular, or neither. Given two lines, they should be able to prove which of these three categories is applicable. They should also know how to find the equation of a line that is parallel or perpendicular to a given line.

Your students may need a quick refresher on some slope basics. You know, point your toes together, watch out for trees, and don't eat yellow snow. That kind of stuff.

A line's slope, of course, is the ratio of rise over run. In the slope-intercept form of a line's equation, it is the coefficient on the x term. Positively sloped lines point up, while negatively sloped ones point down. The greater the magnitude of the slope, the steeper the line.

Parallel lines, or what second graders would refer to as "train track lines," have the same slope and never cross each other. Perpendicular lines, on the other hand, cross at a right angle and feature slopes that are opposite reciprocals of each other.

When your students feel confident enough, tell 'em to grab their boots and poles. Time to hit the slopes!

#### Drills

1. Line AB has a slope of -6 and is parallel to line CD. What is the slope of line CD?

-6

Parallel lines always have the same slope. Always. The other choices are incorrect because they are not equal to -6. Sucks for them.

2. Line JK has the slope ¾ and is perpendicular to line LM. What is the slope of line LM?

-43

Perpendicular lines have opposite reciprocal slopes. The other choices are incorrect because the slope of LM must be both negative and the reciprocal of .

3. Line r has slope -0.5 and line s is perpendicular to it. Which of these could be the equations for line s?

y = 2x – 0.5

Perpendicular lines have opposite reciprocal slopes, and the opposite reciprocal of -0.5 is 2. The slope of the line is the coefficient of x, and (C) is the only choice with a slope of 2. The other choices do not have the correct slope.

4. In slope-intercept form, what is the equation of the line parallel to y = 3x + 5 that passes through the point (6, 2)?

y = 3x – 16

The line y = 3x – 16 is parallel to y = 3x + 5 because it has the same slope, 3, and it passes through the given point. While (A) has the correct slope, it has the wrong y-intercept. We know this because substituting x = 6 into the equation doesn't give us the correct y value of 2.

5. In slope-intercept form, what is the equation of the line parallel to y = 4x – 5 that passes through the point (-2, 4)?

y = 4x + 12

We know our line has to have a slope of 4 and go through the given point. While (A) has the correct slope, it has the wrong intercept. It's clear that (B) and (D) have the opposite reciprocal slope. We know (C) works because 4 = 4(-2) + 12 = -8 + 12 = 4.

6. In slope-intercept form, what is the equation of the line parallel to y = -7x – 2 that passes through the point (3, -1)?

y = -7x + 20

Our line must have a slope of -7 and pass through (3, -1). This means we need to solve the equation -1 = -7(3) + b for the y-intercept. We should get b = 20. Options (B) and (C) have the incorrect slope or y-intercept, while (D) switches x and y.

7. In slope-intercept form, what is the equation of a line perpendicular to y = 2x + 7 that passes through the point (5, 8)?

y = -0.5x + 10.5

Our line must have a slope of -0.5, which is the opposite reciprocal of the slope of the given line. This narrows our options down to only (B). The correct intercept is 10.5 because 8 = -0.5(5) + 10.5 = -2.5 + 10.5 = 8.

8. In slope-intercept form, what is the equation of a line perpendicular to y = 3x – 5 that passes through the point (-3, -6)?

y = -⅓x – 7

It has a slope of -⅓, which is the opposite reciprocal of the slope of the given line, 3, and it passes through the point (-3, -6). None of the other choices have the correct slope, though (D) has the correct intercept.

9. In slope-intercept form, what is the equation of a line perpendicular to y = 10x – 8 that passes through the point (0, 0)?

y = -0.1x

Perpendicular means we need a slope that's the opposite reciprocal of 10, which means -0.1. We also know that our y-intercept is 0 because the point (0, 0) is on the y-axis. While (A) and (B) have the right slope, their y-intercepts are incorrect. The y-intercept is correct for (D), but the slope isn't.

10. Line a passes through point (6, 0) and (2, 8). Line b passes through point (0, -3) and (-2, -4). Which description best fits these lines, and how can you tell?

Lines a and b are perpendicular because they have opposite reciprocal slopes.