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3.6 Perpendiculars and Distance3.6 Perpendiculars and Distance

Distance from a Point to a LineDistance from a Point to a Line

The distance from a line to a point not on the line is the length of the segment ┴ to the line from the point.

l

A

Draw the segment that represents the distance from

Since the distance from a line to a point not on the line is the length of the segment perpendicular to the line from the point,

Answer:

Example 1:Example 1:

Turn to Pages 162 – 163 in your textbook and complete #11 – 16.

Your Turn:Your Turn:

Constructing a Constructing a ┴ Segment┴ Segment

How do we construct a ┴ segment accurately?

How do we use a compass?

By using a compass.

The next example will show us.

Construct a line perpendicular to line s through V(1, 5) not on s. Then find the distance from V to s.

Example 2:Example 2:

Graph line s and point V. Place the compass point at point V. Make the setting wide enough so that when an arc is drawn, it intersects s in two places. Label these points of intersection A and B.

Example 2:Example 2:

Put the compass at point A and draw an arc below line s.

(Hint: Any compass setting greater than will work.)

Example 2:Example 2:

Using the same compass setting, put the compass at point B and draw an arc to intersect the one drawn in step 2. Label the point of intersection Q.

Example 2:Example 2:

Draw . and s. Use the slopes of and s to verify that the lines are perpendicular.

Example 2:Example 2:

Answer: The distance between V and s is about 4.24 units.

The segment constructed from point V(1, 5) perpendicular to the line s, appears to intersect line s at R(–2, 2). Use the Distance Formula to find the distance between point V and line s.

Example 2:Example 2:

Turn to Page 163 in your textbook and complete #17 – 18.

Your Turn:Your Turn:

Distance Between Parallel LinesDistance Between Parallel Lines

Two lines in a plane are || if they are equidistant everywhere.

To verify if two lines are equidistant find the distance between the two || lines by calculating the distance between one of the lines and any point on the other line.

Theorem 3.9Theorem 3.9

In a plane, if two lines are equidistant from a third line, then the two lines are || to each other.

You will need to solve a system of equations to find the endpoints of a segment that is perpendicular to both a and b. The slope of lines a and b is 2.

Find the distance between the parallel lines a and b whose equations are and respectively.

Example 3:Example 3:

Point-slope form

Add 3 to each side.

First, write an equation of a line p perpendicular to a and b. The slope of p is the opposite reciprocal of 2,

Simplify.

Use the y-intercept of line a, (0, 3), as one of the endpoints of the perpendicular segment.

Example 3:Example 3:

Next, use a system of equations to determine the point of intersection of line b and p.

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

Example 3:Example 3:

Substitute 2.4 for x in the equation for p.

Simplify on each side.

The point of intersection is (2.4, 1.8).

Group like terms on each side.

Example 3:Example 3:

Distance Formula

Then, use the Distance Formula to determine the distance between (0, 3) and (2.4, 1.8).

Answer: The distance between the lines is or about 2.7 units.

Example 3:Example 3:

Turn to Page 163 in your textbook and

complete #19 – 22.

Pre-AP: Add #24

Your Turn:Your Turn: