then/now you proved that two lines are parallel using angle relationships. find the distance between...

You proved that two lines are parallel using angle relationships.

• Find the distance between a point and a line.

• Find the distance between parallel lines.

• equidistant

Distance Between Parallel Lines

Find the distance between the parallel lines a and b whose equations are y = 2x + 3 and y = 2x – 1, respectively.

You will need to solve a system of equations to find the endpoints of a segment that is perpendicular to both a and b. From their equations,we know that the slope of line a and line b is 2.

Sketch line p through they-intercept of line b, (0, –1),perpendicular to lines a and b.

a b

p

Distance Between Parallel Lines

Step 1

Use the y-intercept of line b, (0, –1), as one of the endpoints of the perpendicular segment.

Write an equation for line p. The slope of p is the

opposite reciprocal of

Point-slope form

Simplify.

Subtract 1 from each side.

Distance Between Parallel Lines

Use a system of equations to determine the point of intersection of the lines a and p.

Step 2

Substitute 2x + 3 for y in the second equation.

Group like terms on each side.

Distance Between Parallel Lines

Simplify on each side.

Multiply each side by .

Substitute for x in the

equation for p.

Distance Between Parallel Lines

Simplify.

The point of intersection is or (–1.6, –0.2).

Distance Between Parallel Lines

Use the Distance Formula to determine the distance between (0, –1) and (–1.6, –0.2).

Step 3

Distance Formula

x2 = –1.6, x1 = 0, y2 = –0.2, y1 = –1

Answer: The distance between the lines is about 1.79 units.

A. 2.13 units

B. 3.16 units

C. 2.85 units

D. 3 units

Find the distance between the parallel lines a and b

whose equations are and ,

respectively.