Perpendiculars and Distance Page 215 You proved that two lines are parallel using angle relationships. Find the distance between a point and a line. Find.

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Perpendiculars and Distance Page 215 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.

Page 215

Construct Distance From Point to a Line CONSTRUCTION A certain roof truss 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.

A.AD B.AB C.CX D.AX KITES Which segment represents the shortest distance from point A to DB?

Step 1Find 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, (x 1, y 1 ) = (0, 0) Simplify. The equation of line s is y = –x.

Step 2Write 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, (x 1, y 1 ) = (1, 5) Simplify. The equation of line t is y = x + 4. Subtract 1 from each side.

Step 3Solve the system of equations to determine the point of intersection. line s:y =–x line t:(+) y =x + 4 2y = 4Add the two equations. y = 2Divide each side by 2. Solve for x. 2 =–xSubstitute 2 for y in the first equation. –2 =xDivide each side by –1. The point of intersection is (–2, 2). Let this point be Z.

Step 4Use the Distance Formula to determine the distance 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.

Page 218 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…

Page 218

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 the y-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. Substitutefor 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 x 2 = –1.6, x 1 = 0, y 2 = –0.2, y 1 = –1 Answer: The distance between the lines is about 1.79 units.

Steps to follow…

3-6 Assignment Page 221, 13, 14, 15, 17, 21, 23