Use slopes to determine whether the opposite sides of quadrilateral ABCD are parallel.

30. A ( 0 , 2 ) , B ( 3 , 4 ) , C ( 2 , 7 ) , D ( − 1 , 5 ) eh open 0 comma 2 close comma b open 3 comma 4 close comma c open 2 comma 7 close comma d . open negative 1 comma 5 close
31. A ( − 3 , 1 ) , B ( 1 , − 2 ) , C ( 0 , − 3 ) , D ( − 4 , 0 ) eh . open negative 3 comma 1 close . comma b . open 1 comma negative 2 close . comma c . open 0 comma negative 3 close . comma d . open negative 4 comma 0 close
32. A(1, 1), B(5, 3), C(7, 1), D(3, 0)
33. A ( 1 , 0 ) , B ( 4 , 0 ) , C ( 3 , − 3 ) , D ( − 1 , − 3 ) eh open 1 comma 0 close comma b open 4 comma 0 close comma c . open 3 comma negative 3 close . comma d open negative 1 comma negative 3 close

34. Reasoning Are opposite sides of hexagon RSTUVW below parallel? Justify your answer.

    

35. Which line is perpendicular to 3 y + 2 x = 12 ? 3 y plus . 2 x equals 12 question mark
    1. 6 x − 4 y = 24 6 x minus 4 y equals 24
    2. 2 x + 3 y = 6 2 x plus . 3 y equals 6
    3. y + 3 x = − 2 y plus . 3 x equals , minus 2
    4. y = − 2 x + 2 y equals , minus , 2 x plus 2

Rewrite each equation in slope-intercept form, if necessary. Then determine whether the lines are perpendicular. Explain.

36. y = − x − 7 y − x = 20 table with 2 rows and 1 column , row1 column 1 , y equals negative x minus 7 , row2 column 1 , y minus x equals 20 , end table

37. y = 3 x = − 2 table with 2 rows and 1 column , row1 column 1 , y equals 3 , row2 column 1 , x equals negative 2 , end table

38. 2 x − 7 y = − 42 4 y = − 7 x − 2 table with 2 rows and 1 column , row1 column 1 , 2 x minus 7 y equals negative 42 , row2 column 1 , 4 y equals negative 7 x minus 2 , end table

Developing Proof Explain why each theorem is true for three lines in the coordinate plane.

39. Theorem 3-7: If two lines are parallel to the same line, then they are parallel to each other.
40. Theorem 3-8: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

41. Rail Trail A community recently converted an old railroad corridor into a recreational trail. The graph shows below a map of the trail on a coordinate grid. They plan to construct a path to connect the trail to a parking lot. The new path will be perpendicular to the recreational trail.

    

    1. Write an equation of the line representing the new path.
    2. What are the coordinates of the point at which the path will meet the recreational trail?
    3. If each grid space is 25 yd by 25 yd, how long is the path to the nearest yard?
42. Reasoning Is a triangle with vertices G(3, 2), H(8, 5), and K(0, 10) a right triangle? Justify your answer.
43. Graphing Calculator A B ↔ modified eh b with left right arrow above  contains points A ( − 3 , 2 ) eh open negative 3 comma 2 close  and B(5, 1). C D ↔ modified c d with left right arrow above  contains points C(2, 7) and D ( 1 , − 1 ) . d open 1 comma negative 1 close .  Use your graphing calculator to find the slope of A B ↔ . modified eh b with left right arrow above , .  Enter the x-coordinates of A and B into the L1 list of your list editor. Enter the y-coordinates into the L2 list. In your stat begin box , stat , end box  CALC menu select LinReg ( a x + b ) . open eh x , plus b close , .  Press enter begin box , enter , end box  to find the slope a. Repeat to find the slope of C D ↔ . modified c d with left right arrow above , .  Are A B ↔ modified eh b with left right arrow above  and C D ↔ modified c d with left right arrow above  parallel, perpendicular, or neither?

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