For example, prove or disprove that a figure defined by
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Transcript For example, prove or disprove that a figure defined by
Review for Parallelogram
Properties Quiz
G.GPE.B.4 Use coordinates to prove simple
geometric theorems algebraically. For example,
prove or disprove that a figure defined by four given
points in the coordinate plane is a rectangle.
Warm-up
Show
that A(2, -1), B(1, 3), C(6, 5),
and D(7, 1) are the vertices of a
parallelogram.
Method 1
Show that opposite sides have the same slope, so they are
parallel.
Slope of
AB
Slope of
CD
Slope of
BC
Slope of
DA
3 ( 1)
12
15
76
53
6 1
4
1 1
27
4
2
5
2
5
AB and CD have the same slope so they are parallel.
Similarly,
BC
DA . Because opposite sides are parallel,
ABCD is a parallelogram.
Method 2
Show that opposite sides have the same length.
AB
(1 2 ) 3 ( 1)
CD
( 7 6 ) (1 5 )
17
BC
( 6 1) ( 5 3)
29
DA
( 2 7 ) ( 1 1)
2
2
2
2
2
2
2
2
17
29
AB CD and BC DA . Because both pairs of opposite
sides are congruent,
ABCD is a parallelogram.
Method 3
Show that one pair of opposite sides is congruent and parallel.
Find the slopes and lengths of
Slope of
AB Slope of
AB CD
AB and CD as shown in Methods 1 and 2.
CD 4
17
AB and CD are congruent and parallel,
so ABCD is a parallelogram.
Sometimes, Always or
Never True
The diagonals of a parallelogram are congruent.
Sometimes,
if it’s a
rectangle.
The consecutive angles of a rectangle are congruent
and supplementary.
Always
The diagonals of a rectangle bisect each other.
Always
The diagonals of a rectangle bisect the angles.
Sometimes, if it’s
a square.
The diagonals of a square are perpendicular
bisectors of eachAlways
other.
A square is a rectangle.
Always
Sometimes, Always or
Never True
A diagonal divides a square into two isosceles right
triangles.
Always
Consecutive angles in a parallelogram are congruent
Sometimes, if it’s a square or a rectangle. Consecutive angles
are always supplementary.
B(1, 5) O(9, 9) X(11, 5) Y(3, 1)
Is
BOXY a rectangle? Why?
The
diagonals of BOXY are
__________________
More Practice!!!
1.
D
C
2.
D
C
2x - 7
A
x+9
B
Perimeter ABCD = 46
Find the measure of
16
7
AB __________,
BC _______________.
4x +3
A
63
B
Perimeter ABCD = 16x – 12.
75
Measure of AD =_____________
READY FOR A
CHALLENGE???
Find the coordinates of point M in parallelogram PRAM
(b-a, c)
M(?, ?)
P
A(b, c)
R(a, 0)