Geometry

2-3 Parallel and perpendicular lines.

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Geometry

2-3 Parallel and perpendicular lines.

Warm UpFind the reciprocal.

1. 2 2.

3.

Find the slope of the line that passes througheach pair of points.4. (2, 2) and (–1, 3)

5. (3, 4) and (4, 6)

6. (5, 1) and (0, 0)

3

2

Geometry

2-3 Parallel and perpendicular lines.

Vocabularyparallel linesperpendicular lines

Geometry

2-3 Parallel and perpendicular lines.

The slope of a line in a coordinate plane is a Ratio that describes the steepness (Rise over Run) of the line. Any two points on a line can be used to determine the slope.

Geometry

2-3 Parallel and perpendicular lines.

Geometry

2-3 Parallel and perpendicular lines.

One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour.

Geometry

2-3 Parallel and perpendicular lines.

Example 1: Finding the Slope of a Line

Use the slope formula to determine the slope the line. CD

Substitute (4, 2) for (x1, y1) and

(–2, 1) for (x2, y2) in the slope formula and then simplify.

Geometry

2-3 Parallel and perpendicular lines.

TEACH! Example 1

Use the slope formula to determine the

slope of JK through J(3, 1) and K(2, –1).

Substitute (3, 1) for (x1, y1) and (2, –1) for (x2, y2) in the slope formula and then simplify.

Geometry

2-3 Parallel and perpendicular lines.

TEACH: Example 1a

Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither.

WX and YZ for W(3, 1), X(3, –2), Y(–2, 3), and Z(4, 3)

Vertical and horizontal lines are perpendicular.

Geometry

2-3 Parallel and perpendicular lines.

TEACH: Example 1c Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither.

BC and DE for B(1, 1), C(3, 5), D(–2, –6), and E(3, 4)

The lines have the same slope, so they are parallel.

Geometry

2-3 Parallel and perpendicular lines.

Geometry

2-3 Parallel and perpendicular lines.

Geometry

2-3 Parallel and perpendicular lines.

In a parallelogram, opposite sides are parallel.

Remember!

Geometry

2-3 Parallel and perpendicular lines.

Example 2:Show that JKLM is a parallelogram.

Use the ordered pairs and the slope formula to find the slopes of MJ and KL.

MJ is parallel to KL because they have the same slope.

JK is parallel to ML because they are both horizontal.

Since opposite sides are parallel, JKLM is a parallelogram.

Geometry

2-3 Parallel and perpendicular lines.

TEACH! Example 2Show that the points A(0, 2), B(4, 2), C(1, –3), D(–3, –3) are the vertices of a parallelogram.

AD is parallel to BC because they have the same slope.

• •

••A(0, 2)B(4, 2)

D(–3, –3) C(1, –3)

AB is parallel to DC because they are both horizontal. Since opposite sides are parallel, ABCD is a parallelogram.

Use the ordered pairs and slope formula to find the slopes of AD and BC.

Geometry

2-3 Parallel and perpendicular lines.

Perpendicular lines are lines that intersect to form right angles (90°).

Geometry

2-3 Parallel and perpendicular lines.

Identify which lines are perpendicular: y = 3; x = –2; y = 3x; .

The graph given by y = 3 is a horizontal line, and the graph given by x = –2 is a vertical line. These lines are perpendicular.

y = 3x = –2

y =3x

Example 3: Identifying Perpendicular Lines

Geometry

2-3 Parallel and perpendicular lines.

y = 3x = –2

y =3x

Continue

These lines are perpendicular because the product of their slopes is –1.

The slope of the line given by y = 3x is 3. The slope of the line described by is .

Geometry

2-3 Parallel and perpendicular lines.

TEACH! Example 3Identify which lines are perpendicular: y = –4;

y – 6 = 5(x + 4); x = 3; y = The graph described by x = 3 is a vertical line, and the graph described by y = –4 is a horizontal line. These lines are perpendicular.

The slope of the line described by y – 6 = 5(x + 4) is 5. The slope of the line described by y = is

y = –4

x = 3

y – 6 = 5(x + 4)

Geometry

2-3 Parallel and perpendicular lines.

TEACH! Example 3 ContinuedIdentify which lines are perpendicular: y = –4; y – 6 = 5(x + 4); x = 3; y =

These lines are perpendicular because the product of their slopes is –1.

y = –4

x = 3

y – 6 = 5(x + 4)

Geometry

2-3 Parallel and perpendicular lines.

Slopes of Parallel and Perpendicular Lines

•The slopes of two nonvertical lines are equal.

•Two lines with the same slope are parallel

•Vertical lines are parallel

•The product of the slopes of two perpendicularlines, neither of which is vertical, is -1.

•If the product of the slopes of two lines is -1, then the two lines are perpendicular

•A horizontal line and a vertical line are perpendicular.

Geometry

2-3 Parallel and perpendicular lines.

Helpful Hint

If you know the slope of a line, the slope of a perpendicular line will be the "opposite reciprocal.”

Geometry

2-3 Parallel and perpendicular lines.

Facts: Theorems

•Two lines parallel to a third line are parallel to each other.

•In a plane, two lines perpendicular to a third line are parallel to each other.

Geometry

2-3 Parallel and perpendicular lines.

Example 4:

Show that ABC is a right triangle.

slope of

slope of

Therefore, ABC is a right triangle because it contains a right angle.

If ABC is a right triangle, AB will be perpendicular to AC.

AB is perpendicular to AC because

Geometry

2-3 Parallel and perpendicular lines.

TEACH! Example 4Show that P(1, 4), Q(2, 6), and R(7, 1) are the vertices of a right triangle.

PQ is perpendicular to PR because the product of their slopes is –1.

slope of PQ

Therefore, PQR is a right triangle because it contains a right angle.

If PQR is a right triangle, PQ will be perpendicular to PR.

P(1, 4)

Q(2, 6)

R(7, 1) slope of PR

Geometry

2-3 Parallel and perpendicular lines.

Lesson Quiz: Part I

Write an equation in slope-intercept form for the line described.

1. contains the point (8, –12) and is parallel to

2. contains the point (4, –3) and is perpendicular

to y = 4x + 5

Geometry

2-3 Parallel and perpendicular lines.

Lesson Quiz: Part II

3. Show that WXYZ is a rectangle.

The product of the slopes of adjacent sides is –1. Therefore, all angles are right angles, and WXYZ is a rectangle.

slope of = XY

slope of YZ = 4

slope of = WZ

slope of XW = 4