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Lesson Menu Five-Minute Check (over Lesson 3–5) NGSSS Then/Now New Vocabulary Key Concept: Distance Between a Point and a Line Postulate 3.6: Perpendicular Postulate Example 1:Real-World Example: Construct Distance From Point to a Line Example 2:Distance from a Point to a Line on Coordinate Plane Key Concept: Distance Between Parallel Lines Theorem 3.9: Two Line Equidistant from a Third Example 3:Distance Between Parallel Lines

Over Lesson 3–5 A.A B.B C.C D.D 5-Minute Check 1 Given  9   13, which segments are parallel? A.AB || CD B.FG || HI C.CD || FG D.none ___ __

Over Lesson 3–5 A.A B.B C.C D.D 5-Minute Check 2 A.AB || CD B.CD || FG C.FG || HI D.none Given  2   5, which segments are parallel? A.AB || CD B.CD || FG C.FG || HI D.none ___ __

Over Lesson 3–5 A.A B.B C.C D.D 5-Minute Check 3 A.If consecutive interior  s are supplementary, lines are ||. B.If alternate interior  s are , lines are ||. C.If corresponding  s are , lines are ||. D.If 2 lines cut by a transversal so that corresponding  s are , then the lines are ||. If m  2 + m  4 = 180, then AB || CD. What postulate supports this? ___

Over Lesson 3–5 A.A B.B C.C D.D 5-Minute Check 4 If  5   14, then CD || HI. What postulate supports this? __ ___ A.If corresponding  s are , lines are ||. B.If 2 lines are to the same line, they are ||. C.If alternate interior  s are , lines are ||. D.If consecutive interior  s are supplementary, lines are ||. ┴

Over Lesson 3–5 A.A B.B C.C D.D 5-Minute Check 5 A.6.27 B.11 C D.18. Find x so that AB || HI if m  1 = 4x + 6 and m  14 = 7x – 27. __ ___

Over Lesson 3–5 Two lines in the same plane do not intersect. Which term best describes the relationship between the lines? A.A B.B C.C D.D 5-Minute Check 6 A.parallel B.perpendicular C.skew D.transversal

NGSSS MA.912.G.1.2 Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass or a drawing program, explaining and justifying the process used.

Then/Now You proved that two lines are parallel using angle relationships. (Lesson 3–3) Find the distance between a point and a line. Find the distance between parallel lines.

Vocabulary equidistant

Concept

Example 1 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.

Example 1 Construct Distance From Point to a Line 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 AB represents the shortest length of wood needed to connect the peak of the roof to the main beam. ___

A.A B.B C.C D.D Example 1 A.AD B.AB C.CX D.AX KITES Which segment shows the shortest distance from point A to DB?

Step 1Find the slope for 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). Example 2 Distance from a Point to a Line on Coordinate Plane

Example 2 Distance from a Point to a Line on Coordinate Plane 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.

Example 2 Distance from a Point to a Line on Coordinate Plane 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.

Example 2 Distance from a Point to a Line on Coordinate Plane 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 in (–2, 2). Let this point be Z.

Example 2 Distance from a Point to a Line on Coordinate Plane 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.

A.A B.B C.C D.D Example 2 COORDINATE GEOMETRY Line n contains points (2, 4) and (–4, –2). Find the distance between line n and point B(3, 1). A. B. C. D.

Concept

Example 3 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 the y-intercept of line b, (0, –1), perpendicular to lines a and b. a b p

Example 3 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.

Example 3 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.

Example 3 Distance Between Parallel Lines Simplify on each side. Multiply each side by. Substitutefor x in the equation for p.

Example 3 Distance Between Parallel Lines Simplify. The point of intersection is or (–1.6, –0.2).

Example 3 Distance Between Parallel Lines 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.

A.A B.B C.C D.D Example 3 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.

End of the Lesson