Page 580 - Algebra 1

SOLUTION Use the distance formula to find the length of each side of ABCD.
AB= √(x2 -x1)2 +(y2 -y1)2
= √(2-(-1))2 +(1-0)2
BC= √(x2 -x1)2 +(y2 -y1)2
= √(2-1)2 +(1-(-2))2
= √ 3 + 1 = √9 + 1 = √ 1 0
CD= √(x2 -x1)2 +(y2 -y1)2
= √(1-(-2))2 +(-2-(-3))2
= √ 1 + 3 = √1 + 9 = √ 1 0
AD= √(x2 -x1)2 +(y2 -y1)2
= √(-1-(-2))2 +(0-(-3))2
22 22
= √ 3 + 1 = √9 + 1 = √ 1 0
= √ 1 + 3 = √1 + 9 = √ 1 0
22 22
ABCD is a quadrilateral with four congruent sides, so ABCD is a rhombus.
The midpoint of a line segment is the point that divides the segment into two equal-length segments. You can find the coordinates of the midpoint of a line segment by using the midpoint formula.
Math Reasoning
Connect How is finding a midpoint related to finding an average?
The Midpoint Formula
The midpoint M of the line segment y w i t h e n d p o i n_t s ( x , y _) a n d ( x , y ) i s
(x+x y+y)
(x2, y2)
M= x1+x2,y1+y2
11 22
__
12
M=2,2. 22
12
(x1, y1)
x
()
Example
4
Finding the Midpoint of a Segment
Find the midpoint of the line segment with the given endpoints. (3, 5) and (7, -2)
SOLUTION Use the midpoint formula. Substitute (3, 5) f_or (x , y_) and (7, -2) for (x , y ).
Math Reasoning
Generalize Does it make a difference as to which point is (x1, y1) or (x2, y2)? Explain.
11 22
8
(3,
5)
12 M = (x + x , y + y )
= 3 + 7, 5 + (-2) _2 _2
12
4
O
m
x
22 __
Substitute.
Simplify.
S i m p l i f y .
-4
4
8
(7, -2)
= (10 , 3 ) 22
= ( 5 , _3 ) 2
The midpoint of the line segment with endpoints (3, 5) and (7, -2) is (5, _3 ). 2
-8
y
Lesson 86 565