a To classify ABCD, find the slope of each side of it.
B
b To classify ABCD, find the slope of each side of it.
C
c Compare the side lengths of the quadrilaterals.
A
a Rectangle
B
b Rectangle
C
c Yes, see solution.
Practice makes perfect
a Let's examine the given diagram.
There can be more than one way to classify ABCD. One way is to classify by the slope of each side. To do so, we will use the Slope Formula.
Side
Points
y_2-y_1/x_2-x_1
Slope
AB
A( -4, 4) and B( -1, -2)
-2- 4/-1-( -4)
-2
BC
B( -1, -2) and C( -3, -3)
-3-( -2)/-3-( -1)
1/2
CD
C( -3, -3) and D( -6, 3)
3-( -3)/-6-( -3)
-2
AD
A( -4, 4) and D( -6, 3)
3- 4/-6-( -4)
1/2
Notice that the opposite sides are parallel and that the consecutive sides are perpendicular. This means that ABCD is a parallelogram with four right angles, so it is a rectangle.
b To classify EFGH, we will follow the same steps as we did in Part A.
Side
Points
y_2-y_1/x_2-x_1
Slope
EF
E( 1, -3) and F( 4, 3)
3-( -3)/4- 1
2
FG
F( 4, 3) and G( 6, 2)
2-( 3)/6-( 4)
-1/2
GH
G( 6, 2) and H( 3, -4)
-4- 2/3- 6
2
EH
E( 1, -3) and H( 3, -4)
-4-( -3)/3- 1
-1/2
Because of the same reasons that we stated in Part A, EFGH is also a rectangle.
c To determine whether ABCD and EFGH are congruent, we will find their side lengths by the Distance Formula.
ABCD
EFGH
Side
Points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
Length
Side
Points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
Length
AB
A( -4, 4) and B( -1, -2)
sqrt(( -1-( -4))^2+( -2- 4)^2)
s sqrt(45)
EF
E( 1, -3) and F( 4, 3)
sqrt(( 4- 1)^2+( 3- 1)^2)
sqrt(45)
BC
B( -1, -2) and C( -3, -3)
sqrt(( -3-( -1))^2+( -3-( -2))^2)
s sqrt(5)
FG
F( 4, 3) and G( 6, 2)
sqrt(( 6- 4)^2+( 2- 3)^2)
sqrt(5)
CD
C( -3, -3) and D( -6, 3)
sqrt(( -6-( -3))^2+( 3-( -3))^2)
sqrt(45)
GH
G( 6, 2) and H( 3, -4)
sqrt(( 3- 6)^2+( -4- 2)^2)
sqrt(45)
AD
A( -4, 4) and D( -6, 3)
sqrt(( -6-( -4))^2+( 3- 4)^2)
sqrt(5)
EH
E( 1, -3) and H( 3, -4)
sqrt(( 3- 1)^2+( -4-( -3))^2)
sqrt(5)
Looking at the table above, we see that the lengths and widths of the rectangles are congruent. Therefore, ABCD and EFGH are congruent.
