6-7 Polygons in the Coordinate Plane

Quick Review

To determine whether sides or diagonals are congruent, use the Distance Formula. To determine the coordinate of the midpoint of a side, or whether the diagonals bisect each other, use the Midpoint Formula. To determine whether opposite sides are parallel, or whether diagonals or sides are perpendicular, use the Slope Formula.

Example

Δ XYZ cap delta  has vertices X(1, 0), Y ( − 2 , − 4 y open negative 2 comma . negative 4 ), and Z ( 4 , − 4 ) . z open 4 comma negative 4 close .  Is Δ XYZ cap delta  scalene, isosceles, or equilateral?

To find the lengths of the legs, use the Distance Formula.

X Y = ( − 2 − 1 ) 2 + ( − 4 − 0 ) 2 = 9 + 16 = 5 Y Z = ( 4 − ( − 2 ) ) 2 + ( − 4 − ( − 4 ) ) 2 = 36 + 0 = 6 X Z = ( 4 − 1 ) 2 + ( − 4 − 0 ) 2 = 9 + 16 = 5 table with 3 rows and 2 columns , row1 column 1 , x y , column 2 equals . square root of open , negative 2 minus 1 , close squared . plus . open , negative 4 minus 0 , close squared end root . equals , square root of 9 plus 16 end root , equals 5 , row2 column 1 , y z , column 2 equals . square root of open . 4 minus . open , negative 2 , close . close squared . plus . open . negative 4 minus . open , negative 4 , close . close squared end root . equals , square root of 36 plus 0 end root , equals 6 , row3 column 1 , x z , column 2 equals . square root of open , 4 minus 1 , close squared . plus . open , negative 4 minus 0 , close squared end root . equals , square root of 9 plus 16 end root , equals 5 , end table

Two side lengths are equal, so Δ XYZ cap delta  is isosceles.

Exercises

Determine whether Δ ABC cap delta  is scalene, isosceles, or equilateral.

What is the most precise classification of the quadrilateral?

40. G ( 2 , 5 ) , R ( 5 , 8 ) , A ( − 2 , 12 ) , D ( − 5 , 9 ) g open 2 comma 5 close comma r open 5 comma 8 close comma . eh open negative 2 comma . 12 close comma . d open negative 5 comma . 9 close
41. F ( − 13 , 7 ) , I ( 1 , 12 ) , N ( 15 , 7 ) , E ( 1 , − 5 ) f open negative 13 comma . 7 close comma i open 1 comma 12 close comma n open 15 comma 7 close comma e open 1 comma , negative 5 , close
42. Q ( 4 , 5 ) , U ( 12 , 14 ) , A ( 20 , 5 ) , D ( 12 , − 4 ) q open 4 comma 5 close comma u open 12 comma 14 close comma eh open 20 comma 5 close comma d open 12 comma , negative 4 , close
43. W ( − 11 , 4 ) , H ( − 9 , 10 ) , A ( 2 , 10 ) , T ( 4 , 4 ) w open negative 11 comma . 4 close comma . h open negative 9 comma . 10 close comma eh open 2 comma 10 close comma t open 4 comma 4 close

6-8 and 6-9 Coordinate Geometry and Coordinate Proofs

Quick Review

When placing a figure in the coordinate plane, it is usually helpful to place at least one side on an axis. Use variables when naming the coordinates of a figure in order to show that relationships are true for a general case.

Example

Rectangle PQRS has length a and width 4b. The x-axis bisects P S ¯ p s bar  and Q R ¯ . q r bar , .  What are the coordinates of the vertices?

Since the width of PQRS is 4b and the x-axis bisects P S ¯ p s bar  and Q R ¯ , q r bar , comma  all the vertices are 2b units from the x-axis. P S ¯ p s bar  is on the y-axis, so P = ( 0 , 2 b ) p equals open 0 comma 2 b close  and S = ( 0 , − 2 b ) . s equals open 0 comma . negative 2 b close .  The length of PQRS is a, so Q = ( a , 2 b ) q equals open eh comma 2 b close  and R = ( a , − 2 b ) . r equals open eh comma . negative 2 b close .

Exercises

44. In rhombus FLPS, the axes form the diagonals. If S L = 2 a s l equals 2 eh  and F P = 4 b , f p equals 4 b comma  what are the coordinates of the vertices?

    

45. The figure below is a parallelogram. Give the coordinates of point P without using any new variables.

    

46. Use coordinate geometry to prove that the quadrilateral formed by connecting the midpoints of a kite is a rectangle.

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