1-7 Midpoint and Distance in the Coordinate Plane

Quick Review

You can find the coordinates of the midpoint M of A B ¯ eh b bar  with endpoints A ( x 1 , y 1 ) eh . open , x sub 1 , comma , y sub 1 , close  and B ( x 2 , y 2 ) b . open , x sub 2 , comma , y sub 2 , close  using the Midpoint Formula.

M ( x 1 + x 2 2 , y 1 + y 2 2 ) m . open . fraction x sub 1 , plus , x sub 2 , over 2 end fraction . comma . fraction y sub 1 , plus , y sub 2 , over 2 end fraction . close

You can find the distance d between two points A ( x 1 , y 1 ) eh . open , x sub 1 , comma , y sub 1 , close  and B ( x 2 , y 2 ) b . open , x sub 2 , comma , y sub 2 , close  using the Distance Formula.

d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d equals . square root of open . x sub 2 , minus , x sub 1 . close squared . plus . open . y sub 2 , minus , y sub 1 . close squared end root

Example

G H ¯ g h bar  has endpoints G ( − 11 , 6 ) g open negative 11 comma 6 close  and H(3, 4). What are the coordinates of its midpoint M?

x -coordinate = − 11 + 3 2 = − 4 y -coordinate = 6 + 4 2 = 5 table with 2 rows and 1 column , row1 column 1 , x . minuscoordinate . equals . fraction negative 11 plus 3 , over 2 end fraction . equals negative 4 , row2 column 1 , y . minuscoordinate . equals . fraction 6 plus 4 , over 2 end fraction . equals 5 , end table

The coordinates of the midpoint of G H ¯ g h bar  are M ( − 4 , 5 ) . m open negative 4 comma 5 close .

Exercises

Find the distance between the points to the nearest tenth.

30. A ( − 1 , 5 ) , B ( 0 , 4 ) eh open negative 1 comma 5 close comma b open 0 comma 4 close
31. C ( − 1 , − 1 ) , D ( 6 , 2 ) c open negative 1 comma negative 1 close comma d open 6 comma 2 close
32. E ( − 7 , 0 ) , F ( 5 , 8 ) e open negative 7 comma 0 close comma f open 5 comma 8 close

A B ¯ eh b bar  has endpoints A ( − 3 , 2 ) eh open negative 3 comma 2 close  and B ( 3 , − 2 ) . b open 3 comma negative 2 close .

33. Find the coordinates of the midpoint of A B ¯ . eh b bar , .
34. Find AB to the nearest tenth.

M is the midpoint of J K ¯ . j k bar , .  Find the coordinates of K.

35. J ( − 8 , 4 ) , M ( − 1 , 1 ) j open negative 8 comma 4 close comma m open negative 1 comma 1 close
36. J ( 9 , − 5 ) , M ( 5 , − 2 ) j open 9 comma negative 5 close comma m open 5 comma negative 2 close
37. J ( 0 , 11 ) , M ( − 3 , 2 ) j open 0 comma 11 close comma m open negative 3 comma 2 close

1-8 Perimeter, Circumference, and Area

Quick Review

The perimeter P of a polygon is the sum of the lengths of its sides. Circles have a circumference C. The area A of a polygon or a circle is the number of square units it encloses.

• Square: P = 4 s ; A = s 2 p equals 4 s semicolon eh equals , s squared
• Rectangle: P = 2 b + 2 h ; A = b h p equals 2 b plus 2 h semicolon eh equals b h
• Triangle: P = a + b + c ; A = 1 2 b h p equals eh plus b plus c semicolon eh equals , 1 half , b h
• Circle: C = π d  or  C = 2 π r ; A = π r 2 c equals pi d , or c equals 2 pi r semicolon eh equals pi , r squared

Example

Find the perimeter and area of a rectangle with b = 12 bold italic b equals 12  m and h = 8 bold italic h equals 8  m.

P = 2 b + 2 h A = b h   = 2 ( 12 ) + 2 ( 8 )   = 12 · 8   = 40   = 96 table with 3 rows and 4 columns , row1 column 1 , p , column 2 equals 2 b plus 2 h , column 3 eh , column 4 equals b h , row2 column 1 , , column 2 equals 2 open 12 close plus 2 open 8 close , column 3 , column 4 equals 12 middle dot 8 , row3 column 1 , , column 2 equals 40 , column 3 , column 4 equals 96 , end table

The perimeter is 40 m and the area is 96   m 2 . 96 , m squared , .

Exercises

Find the perimeter and area of each figure.

Find the circumference and the area for each circle in terms of π.

40. r = 3  in. r equals 3 , in.
41. d = 15  m d equals 15 m

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