Perpendiculars and Distance

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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.

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Construct Distance From Point to a LineCONSTRUCTION A certain rooftruss is designed so that the center post extends from the peak of the roof (point A) to the main beam. Construct and name the segment whose length represents the shortest length of wood that will be needed to connect the peak of the roof to the main beam.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. Locate points R and S on the main beam equidistant from point A.

Locate a second point not on the beam equidistant from R and S. Construct AB so that AB is perpendicular to the beam.

Answer: The measure of represents the shortest length of wood needed to connect the peak of the roof to the main beam.

A. AD

B. AB

C. CX

D. AX

KITES Which segment represents the shortest distance from point A to DB?

Step 1 Find the slope of line s.Begin by finding the slope of the line through points (0, 0) and (–5, 5).

COORDINATE GEOMETRY Line s contains points at (0, 0) and (–5, 5). Find the distance between line s and point V(1, 5).

(–5, 5)

(0, 0)

V(1, 5)

Then write the equation of this line by using the point (0, 0) on the line.

Slope-intercept form

m = –1, (x1, y1) = (0, 0)Simplify.

The equation of line s is y = –x.

Step 2 Write an equation of the line t perpendicular to line s through V(1, 5).

Since the slope of line s is –1, the slope of line t is 1. Write the equation for line t through V(1, 5) with a slope of 1.

Slope-intercept form

m = 1, (x1, y1) = (1, 5)

Simplify.

The equation of line t is y = x + 4.

Subtract 1 from each side.

Step 3 Solve the system of equations to determine the point of intersection.

line s: y = –xline t: (+) y = x + 4

2y = 4 Add the two equations.y = 2 Divide each side by 2.

Solve for x.2 = –x Substitute 2 for y in the first

equation.–2 = x Divide each side by –1.

The point of intersection is (–2, 2). Let this point be Z.

Step 4 Use the Distance Formula to determine thedistance between Z(–2, 2) and V(1, 5).

Distance formula

Substitution

Simplify.

Answer: The distance between the point and the line is or about 4.24 units.

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By definition, parallel lines do not intersect. An alternate definition states that two lines in a plane are parallel if they are everywhere equidistant.Equidistant means that the distance between two lines measured along a perpendicular line to the lines is always the same.

Steps to follow…1. Find the slope2. Write the equation of the line.3. Write the equation of a line perpendicular

through the point given.4. Take equations from #2 and #3 and solve for

x and y using elimination.5. Use the distance formula to find the distance

between the given point and the point you found in #4.

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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

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.

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

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

Group like terms on each side.

Simplify on each side.

Multiply each side by .

Substitute for x in the

equation for p. Simplify.

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.

Steps to follow…1. Find the slope2. Write the equation of the line.3. Write the equation of a line perpendicular

through the point given.4. Take equations from #2 and #3 and solve for

x and y using elimination.5. Use the distance formula to find the

distance between the given point and the point you found in #4.

3-6 Assignment

Page 221, 13, 14, 15, 17, 21, 23