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 A B C D

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Given 9 13, which segments are parallel?

A. AB || CD

B. FG || HI

C. CD || FG

D. none

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Over Lesson 3–5

A. A

B. B

C. C

D. D A B C D

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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

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Over Lesson 3–5

A. A

B. B

C. C

D. D A B C D

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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 m2 + m4 = 180, then AB || CD. What postulate supports this?

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Over Lesson 3–5

A. A

B. B

C. C

D. D A B C D

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If 5 14, then CD || HI. What postulate supports this?

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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 ||.

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Over Lesson 3–5

A. A

B. B

C. C

D. D A B C D

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A. 6.27

B. 11

C. 14.45

D. 18

. Find x so that AB || HI if m1 = 4x + 6 and m14 = 7x – 27.

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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 A B C D

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A. parallel

B. perpendicular

C. skew

D. transversal

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.

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.

• equidistant

Construct Distance From Point to a Line

CONSTRUCTION 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.

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.

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A. A

B. B

C. C

D. D A B C D

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A. AD

B. AB

C. CX

D. AX

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

Step 1 Find 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).

Distance from a Point to a Line on Coordinate Plane

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, (x1, y1) = (0, 0)

Simplify.

The equation of line s is y = –x.

Distance from a Point to a Line on Coordinate Plane

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.

Distance from a Point to a Line on Coordinate Plane

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

line s: y = –x

line 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 in (–2, 2). Let this point be Z.

Distance from a Point to a Line on Coordinate Plane

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.

A. A

B. B

C. C

D. D A B C D

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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.

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 they-intercept of line b, (0, –1),perpendicular to lines a and b.

a b

p

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.

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.

Distance Between Parallel Lines

Simplify on each side.

Multiply each side by .

Substitute for x in the

equation for p.

Distance Between Parallel Lines

Simplify.

The point of intersection is or (–1.6, –0.2).

Distance Between Parallel Lines

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.

A. A

B. B

C. C

D. D A B C D

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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.