Perpendiculars and Distance

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Ira Gardner
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erpendiculars and Distance Vocabular equidistant Find the distance between a point and a line. Find the distance between parallel lines.

ne technique is to install the first brace and then use a carpenter s square to measure and mark two or more points the same distance from the first brace.

The carpenter s square is used to construct a line perpendicular to each pair of shelves. The space between each pair of shelves is measured along the perpendicular segment.

The shortest segment from a point to a line is the perpendicular segment from the point to the line.

1 erpendiculars and Distance Vocabular equidistant Find the distance between a point and a line. Find the distance between parallel lines. does the distance between parallel lines relate to hanging new shelves? When installing shelf brackets, it is important that the vertical bracing be parallel in order for the shelves to line up. ne technique is to install the first brace and then use a carpenter s square to measure and mark two or more points the same distance from the first brace. You can then align the second brace with those marks. DISTNE FRM INT T LINE In Lesson -5, ou learned that if two lines are perpendicular to the same line, then the are parallel. The carpenter s square is used to construct a line perpendicular to each pair of shelves. The space between each pair of shelves is measured along the perpendicular segment. This is to ensure that the shelves are parallel. This is an example of using lines and perpendicular segments to determine distance. The shortest segment from a point to a line is the perpendicular segment from the point to the line. Words 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. Distance etween a oint and a Line Model shortest distance Stud Tip Measuring the Shortest Distance You can use tools like the corner of a piece of paper or our book to help draw a right angle. Example 1 Distance from a oint to a Line Draw the segment that represents the distance from to. 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, extend and draw so that. When ou draw a perpendicular segment from a point to a line, ou can guarantee that it is perpendicular b using the construction of a line perpendicular to a line through a point not on that line. Lesson -6 erpendiculars and Distance 159 aron Haupt

2 Example 2 onstruct a erpendicular Segment RDINTE GEMETRY Line contains points and (0, 1). onstruct a line perpendicular to line through ( 7, 2) not on. Then find the distance from to. 1 Graph line and point. lace the compass point at point. Make the setting wide enough so that when an arc is drawn, it intersects in two places. Label these points of intersection and. - (0, 1) x 2 ut the compass at point and draw an arc below line. (Hint: n compass setting greater than 1 2 will work.) - (0, 1) x Stud Tip Distance Note that the distance from a point to the x-axis can be determined b looking at the -coordinate and the distance from a point to the -axis can be determined b looking at the x-coordinate. Using the same compass setting, put the compass at point and draw an arc to intersect the one drawn in step 2. Label the point of intersection. Draw.. Label point R at the intersection of and. Use the slopes of and to verif that the lines are perpendicular. The segment constructed from point ( 7, 2) perpendicular to the line, appears to intersect line at R(, 5). Use the Distance Formula to find the distance between point and line. d (x x 2 1 ) 2 ( 2 1 ) 2 ( 7 ( )) 2 2 ( 5)) 2 - (0, 1) x - (0, 1) x R 25 or 5 The distance between and is 5 units. DISTNE ETWEEN RLLEL LINES Two lines in a plane are parallel if the are everwhere equidistant. Equidistant means that the distance between two lines measured along a perpendicular line to the lines is alwas the same. The distance between parallel lines is the length of the perpendicular segment with endpoints that lie on each of the two lines. D E K J H G F K J H DG EF 160 hapter arallel and erpendicular Lines

3 Distance etween arallel Lines The distance between two parallel lines is the distance between one of the lines and an point on the other line. Stud Tip Look ack To review locus, see Lesson 1-1. Recall that a locus is the set of all points that satisf a given condition. arallel lines can be described as the locus of points in a plane equidistant from a given line. d d Theorem.9 In a plane, if two lines are equidistant from a third line, then the two lines are parallel to each other. Example Distance etween Lines Find the distance between the parallel lines and m whose equations are 1 x and 1 x 1, respectivel. You will need to solve a sstem of equations to find the endpoints of a segment that is perpendicular to both and m. The slope of lines and m is 1. First, write an equation of a line p perpendicular to and m. The slope of p is the opposite reciprocal of 1, or. Use the -intercept of line, (0, ), as one of the endpoints of the perpendicular segment. 1 m(x x 1 ) oint-slope form ( ) (x 0) x 1 0, 1, m x x Simplif. Subtract from each side. Next, use a sstem of equations to determine the point of intersection of line m and p. m: 1 x 1 p: x The point of intersection is (1, 0). Then, use the Distance Formula to determine the distance between (0, ) and (1, 0). d (x x 2 1 ) 2 ( 2 1 ) 2 Distance Formula (0 ) 1 2 ( 0) 2 x 2 = 0, x 1 = 1, 2 =, 1 = x 1 x 1 x x 1 Simplif. The distance between the lines is 10 or about.16 units. (0, ) Substitute 1 x 1 second equation. p m for in the Group like terms on each side. 1 0 x 10 Simplif on each side. x 1 Divide each side b 1. 0 (1) Substitute 1 for x in the equation for p. 0 Simplif. x Lesson -6 erpendiculars and Distance 161

4 oncept heck Guided ractice 1. Explain how to construct a segment between two parallel lines to represent the distance between them. 2. EN ENDED Make up a problem involving an everda situation in which ou need to find the distance between a point and a line or the distance between two lines. For example, find the shortest path from the patio of a house to a garden to minimize the length of a walkwa and material used in its construction.. ompare and contrast three different methods that ou can use to show that two lines in a plane are parallel. op each figure. Draw the segment that represents the distance indicated.. L to KN L M 5. D to E D K N E 6. RDINTE GEMETRY Line contains points (0, 0) and (2, ). Draw line. onstruct a line perpendicular to through (2, 6). Then find the distance from to. Find the distance between each pair of parallel lines. 7. x 1 8. x 6 x 1 8 x 1 pplication 9. Graph the line whose equation is x 1. onstruct a perpendicular segment through (2, 5). Then find the distance from to the line. 10. UTILITIES Housing developers often locate the shortest distance from a house to the water main so that a minimum of pipe is required to connect the house to the water suppl. op the diagram, and draw a possible location for the pipe. connection Water main ractice and ppl For Exercises , See Examples Extra ractice See page 760. op each figure. Draw the segment that represents the distance indicated. 11. to D 12. K to JL 1. to RS J R D L K S 162 hapter arallel and erpendicular Lines

op each figure. Draw the segment that represents the distance indicated. 1. Y to WX 15. G to HJ 16. W to UV W X M G R S Z Y L K J H X W V U T RDINTE GEMETRY onstruct a line perpendicular to through.

18. Line contains points (0, 2) and (1, ). oint has coordinates (, ).

5 op each figure. Draw the segment that represents the distance indicated. 1. Y to WX 15. G to HJ 16. W to UV W X M G R S Z Y L K J H X W V U T RDINTE GEMETRY onstruct a line perpendicular to through. Then find the distance from to. 17. Line contains points (, 0) and (, 0). oint has coordinates (, ). 18. Line contains points (0, 2) and (1, ). oint has coordinates (, ). Interior Designer Interior designers compute areas and volumes, work with scale models, and create budgets, in addition to being artistic. Usuall two ears of college and two ears of practical experience are necessar before a designer can take a licensing exam. nline Research For information about a career as an interior designer, visit: com/careers Find the distance between each pair of parallel lines x 21. 2x 2 1 x 2 2x 22. x 2. 2x 2. x 1 x 17 2x x 20 Graph each line. onstruct a perpendicular segment through the given point. Then find the distance from the point to the line , ( 2, ) 26. 2x 2, ( 1, 5) 27. 2x 9, (2, 0) 28. RF Write a paragraph proof of Theorem INTERIR DESIGN Theresa is installing a curtain rod on the wall above the window. In order to ensure that the rod is parallel to the ceiling, she measures and marks 9 inches below the ceiling in several places. If she installs the rod at these markings centered over the window, how does she know the curtain rod will be parallel to the ceiling? 0. NSTRUTIN When framing a wall during a construction project, carpenters often use a plumb line. plumb line is a string with a weight called a plumb bob attached on one end. The plumb line is suspended from a point and then used to ensure that wall studs are vertical. How does the plumb line help to find the distance from a point to the floor? 1. LGER In the coordinate plane, if a line has equation ax b c, then the distance from a point (x 1, 1 ) is given b ax1 b1 c. Determine the distance from (, 6) to the line a 2 b 2 whose equation is x RITIL THINKING Draw a diagram that represents each description. a. oint is equidistant from two parallel lines. b. oint is equidistant from two intersecting lines. c. oint is equidistant from two parallel planes. d. oint is equidistant from two intersecting planes. e. line is equidistant from two parallel planes. f. plane is equidistant from two other planes that are parallel. Lesson -6 erpendiculars and Distance 16 (l)lonnie Duka/Index Stock Imager/ictureuest, (r)steve henn/ris

Standardized Test ractice. WRITING IN MTH nswer the question that was posed at the beginning of the lesson. How does the distance between parallel lines relate to hanging new shelves?

6 Standardized Test ractice. WRITING IN MTH nswer the question that was posed at the beginning of the lesson. How does the distance between parallel lines relate to hanging new shelves? Include the following in our answer: an explanation of wh marking several points equidistant from the first brace will ensure that the braces are parallel, and a description of other tpes of home improvement projects that require that two or more elements are parallel.. GRID IN Segment is perpendicular to segment D. Segment and segment D bisect each other at point X. If 16 and D 20, what is the length of D? 5. LGER coin was flipped 2 times and came up heads 1 times and tails 10 times. If the first and the last flips were both heads, what is the greatest number of consecutive heads that could have occurred? D 1 Maintain Your Skills Mixed Review Given the following information, determine which lines, if an, are parallel. State the postulate or theorem that justifies our answer. (Lesson -5) and 2 are supplementar. D 5 E F Write an equation in slope-intercept form for each line. (Lesson -) b 9. a 0. b 1. c 2. perpendicular to line a, contains ( 1, ). parallel to line c, contains (2, 5) a x c. RF Write a two-column proof. (Lesson 2-7) Given: NL NM L M rove: N N L M N When Is Weather Normal? It s time to complete our project. Use the information and data ou have gathered about climate and locations on Earth to prepare a portfolio or Web page. e sure to include graphs and/or tables in the presentation hapter arallel and erpendicular Lines

Name Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines

Name Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines Two lines are if they are coplanar and do not intersect. Skew lines. Two