6.2 Slopes of Parallel and Perpendicular Lines

Transcription

1 . Slopes of Parallel and Perpendicular Lines FOCUS Use slope to find out if two lines are parallel or perpendicular. These two lines are parallel. Slope of line AB Slope of line CD These two lines have the same slope. run: A rise: run: C rise: 0 B D In general: Slopes of Parallel Lines Parallel lines have the same slope. Eample 1 Identifing Parallel Lines Line EF passes through E(, ) and F(0, ). Line PQ passes through P(, ) and Q(, ). Line RS passes through R(0, ) and S(, ). Are the lines parallel? Solution F Q S 0 E P R All of them appear to be parallel. Use the formula for slope Copright 011 Pearson Canada Inc.

2 Slope 1 E(, ) and F(0, ) To find the slope of EF,, 1, 0, and () Slope of EF 0 (), or P(, ) and Q(, ) To find the slope of PQ,, 1,, and () Slope of PQ () The slopes of EF and RS are equal, so these two lines are parallel. Although PQ appears to be parallel, it is not. R(0, ) and S(, ) To find the slope of RS, 0, 1,, and Slope of RS () 0, or Check 1. The slope of each line is given. Circle the lines that are parallel. a) Slope AB: b) Slope CD: c) Slope EF: d) Slope GH:. Line AB passes through A(, ) and B( 1, ). Line CD passes through C(, ) and D( 1, ). Line EF passes through E(, 1) and F(, ). Are the lines parallel? Lines and appear to be parallel Copright 011 Pearson Canada Inc.

3 Use the formula: Slope 1 A(, ) and B(1, ) For the slope of AB,, 1,, C(, ) and D(1, ) For the slope of CD,, 1,, E(, 1) and F(, ) For the slope of EF,, 1,, Slope Slope Slope Slope of AB is. Slope of CD is. Slope of EF is. Lines and have slope, so the are. Lines PQ and RS are perpendicular. R Q P S 0 Perpendicular lines intersect at right angles. Slope of PQ Since a 1 b 1, Slope of RS 1 Two numbers are negative reciprocals if their product is 1. then 1 is the negative reciprocal of. In general: Slopes of Oblique Perpendicular Lines The slopes of two oblique perpendicular lines are negative reciprocals; that is, a line with slope m is perpendicular to a line with slope 1 ; m 0. m Recall that an line that is not horizontal or vertical is an oblique line Copright 011 Pearson Canada Inc.

4 Eample Eamining Slopes to Compare Lines Line PQ passes through P( 1, 1) and Q(, ). Line RS passes through R(, ) and S(0, 1). Line TU passes through T(, ) and U(, 1). Are the lines perpendicular? Solution Q T P U 0 S Both PQ and TU appear to be perpendicular to SR. Use the formula. Slope 1 R P(1, 1) and Q(, ) To find the slope of PQ, 1, 1 1,, and (1) Slope of PQ (1) 9 Multipl slopes to check: a 1 1 b R(, ) and S(0, 1) To find the slope of RS,, 1, 0, and 1 Slope of RS 1 () 0 1 T(, ) and U(, 1) To find the slope of TU,, 1,, and 1 Slope of TU 1 The slopes of PQ and RS are negative reciprocals, so these lines are perpendicular Copright 011 Pearson Canada Inc. 7

5 Check 1. The slope of each line is given. Circle the lines that are perpendicular. 1 a) Slope AB: b) Slope CD: c) Slope EF: 1 d) Slope GH: 1. Line PQ passes through P(1, ) and Q(, ). Line RS passes through R( 1, ) and S(, ). Line TU passes through T(, ) and U(, ). Are the lines perpendicular? 10 0 Lines and appear to be perpendicular. Use the formula: Slope 1 P(1, ) and Q(, ) For the slope of PQ,, 1,, R(1, ) and S(, ) For the slope of RS,, 1,, T(, ) and U(, ) For the slope of TU,, 1,, Slope Slope Slope Slope of PQ is. Slope of RS is. Slope of TU is. Multipl slopes to check: PQ and TU: RS and TU: PQ and RS: The slopes of and are, so these lines are Copright 011 Pearson Canada Inc.

6 Eample Using Slope to Identif a Polgon Is quadrilateral BCDE a trapezoid? A trapezoid has one pair of parallel sides. D E B 0 C Solution From the diagram, sides ED and BC are not parallel because the slope of ED is positive and the slope of BC is negative. To check if EB is parallel to DC, compare their slopes. B 1 E D 0 C Slope of EB 1 Slope of DC, or 1 The slopes of EB and DC are equal, so EB and DC are parallel and BCDE is a trapezoid. Check 1. Is ABC a right triangle? A C 0 B Count squares to find each rise and run. From the diagram, looks like a right angle. Find the slopes of sides and to check. Slope of Slope of The slopes negative reciprocals, so sides and perpendicular, and ABC a right triangle Copright 011 Pearson Canada Inc. 9

7 Practice 1. The slopes of lines are given. Circle the slopes of parallel lines. 10 a) Slope: b) Slope: c) Slope: d) Slope:. The slopes of lines are given. Circle the slopes of perpendicular lines. 10 a) Slope: b) Slope: c) Slope: d) Slope: 1. a) Line AB has slope. A line perpendicular to AB has slope: b) Line CD has slope. A line perpendicular to CD has slope: c) Line EF has slope 7. A line perpendicular to EF has slope: d) Line GH has slope 1. A line perpendicular to GH has slope:. Write parallel, perpendicular, or neither to indicate how the lines in each pair are related. 1 1 a) Slope EF: b) Slope MN: c) Slope JK: Slope GH: Slope PQ: 1 Slope MN:. Line AB passes through A(, ) and B(, 0). Line CD passes through C(1, 1) and D(, ). Are the lines parallel? The _ parallel. Use the formula for slope. Slope 1 A(, ) and B(, 0) To find the slope of AB,, 1,, and C(1, 1) and D(, ) To find the slope of CD,, 1,, and Slope of AB Slope of CD The slopes of AB and CD are, so these lines are Copright 011 Pearson Canada Inc.

8 . Line QR passes through Q(, 0) and R(, ). Line ST passes through S(, ) and T(1, ). Are the lines perpendicular? 0 Lines and appear to be. Use the formula: Slope 1 Q(, 0) and R(, ) For the slope of QR,, 1,, Slope S(, ) and T(1, ) For the slope of ST,, 1,, Slope Multipl slopes to check: The slopes of and are, so these lines are Copright 011 Pearson Canada Inc. 1

9 7. Draw quadrilateral ABCD with vertices: A(0, ), B(, ), C(, ) and D(, 1) Is quadrilateral ABCD a parallelogram? 0 From the diagram: Sides and appear to be. Sides and appear to be. To check if is parallel to, compare their slopes. Slope of Slope of The slopes of and are, so and are. To check if is parallel to, compare their slopes. Slope of Slope of The slopes of and are, so and are. Sides and are. Sides and are. The opposite sides of the quadrilateral parallel, so ABCD a parallelogram Copright 011 Pearson Canada Inc.