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SECTION 6-4Rectangles
Monday, April 22, 2013
ESSENTIAL QUESTIONS
How do you recognize and apply properties of rectangles?
How do you determine if parallelograms are rectangles?
Monday, April 22, 2013
RECTANGLE
Monday, April 22, 2013
RECTANGLE
A parallelogram with four right angles.
Monday, April 22, 2013
RECTANGLE
A parallelogram with four right angles.
Four right angles
Monday, April 22, 2013
RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Monday, April 22, 2013
RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Opposite angles are congruent
Monday, April 22, 2013
RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Opposite angles are congruent
Consecutive angles are supplementary
Monday, April 22, 2013
RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
Monday, April 22, 2013
THEOREMS
6.13 - Diagonals of a Rectangle:
6.14 - Diagonals of a Rectangle Converse:
Monday, April 22, 2013
THEOREMS
6.13 - Diagonals of a Rectangle: If a parallelogram is a rectangle, then its diagonals are congruent
6.14 - Diagonals of a Rectangle Converse:
Monday, April 22, 2013
THEOREMS
6.13 - Diagonals of a Rectangle: If a parallelogram is a rectangle, then its diagonals are congruent
6.14 - Diagonals of a Rectangle Converse: If diagonals of a parallelogram are congruent, then the parallelogram is a rectangle
Monday, April 22, 2013
EXAMPLE 1A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet and
LN = 6.5 feet, find KM.
Monday, April 22, 2013
EXAMPLE 1A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet and
LN = 6.5 feet, find KM.
Since we have a rectangle, the diagonals are congruent.
Monday, April 22, 2013
EXAMPLE 1A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet and
LN = 6.5 feet, find KM.
Since we have a rectangle, the diagonals are congruent.
The diagonals also bisect each other, so JN = LN and KN = MN.
Monday, April 22, 2013
EXAMPLE 1A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet and
LN = 6.5 feet, find KM.
Since we have a rectangle, the diagonals are congruent.
The diagonals also bisect each other, so JN = LN and KN = MN.
So JN = LN = KN = MN = 6.5 feet and KM = KN + MN.
Monday, April 22, 2013
EXAMPLE 1A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet and
LN = 6.5 feet, find KM.
Since we have a rectangle, the diagonals are congruent.
The diagonals also bisect each other, so JN = LN and KN = MN.
So JN = LN = KN = MN = 6.5 feet and KM = KN + MN.
KM = 13 feet
Monday, April 22, 2013
EXAMPLE 2Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, find x.
Monday, April 22, 2013
EXAMPLE 2Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, find x.
m∠RTU + m∠SUR = 90
Monday, April 22, 2013
EXAMPLE 2Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, find x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90
Monday, April 22, 2013
EXAMPLE 2Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, find x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90 11x + 2 = 90
Monday, April 22, 2013
EXAMPLE 2Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, find x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90 11x + 2 = 90
−2 −2
Monday, April 22, 2013
EXAMPLE 2Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, find x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90 11x + 2 = 90
−2 −2 11x = 88
Monday, April 22, 2013
EXAMPLE 2Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, find x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90 11x + 2 = 90
−2 −2 11x = 88 11 11
Monday, April 22, 2013
EXAMPLE 2Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, find x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90 11x + 2 = 90
−2 −2 11x = 88 11 11
x = 8
Monday, April 22, 2013
EXAMPLE 3Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a
canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides of the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, and DA = 35 inches, how long do
the lengths of the diagonals need to be?
Monday, April 22, 2013
EXAMPLE 3Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a
canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides of the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, and DA = 35 inches, how long do
the lengths of the diagonals need to be?
The diagonal forms a right triangle with legs of 12 and 35. We need to find
the hypotenuse.
Monday, April 22, 2013
EXAMPLE 3
Monday, April 22, 2013
EXAMPLE 3
a2 + b2 = c2
Monday, April 22, 2013
EXAMPLE 3
a2 + b2 = c2
122 + 352 = c2
Monday, April 22, 2013
EXAMPLE 3
a2 + b2 = c2
122 + 352 = c2
144 + 1225 = c2
Monday, April 22, 2013
EXAMPLE 3
a2 + b2 = c2
122 + 352 = c2
144 + 1225 = c2
1369 = c2
Monday, April 22, 2013
EXAMPLE 3
a2 + b2 = c2
122 + 352 = c2
144 + 1225 = c2
1369 = c2
1369 = c2
Monday, April 22, 2013
EXAMPLE 3
a2 + b2 = c2
122 + 352 = c2
144 + 1225 = c2
1369 = c2
1369 = c2
c = 37
Monday, April 22, 2013
EXAMPLE 3
a2 + b2 = c2
122 + 352 = c2
144 + 1225 = c2
1369 = c2
1369 = c2
c = 37
The diagonals must both be 37 inches
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.Diagonals must be congruent
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2
Diagonals must be congruent
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
Diagonals must be congruent
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25Diagonals must be congruent
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50Diagonals must be congruent
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2
Diagonals must be congruent
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
Diagonals must be congruent
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49
Diagonals must be congruent
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
Diagonals must be congruent
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2 =
13
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2 =
13
m( LM ) = −3 + 2
0− 3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2 =
13
m( LM ) = −3 + 2
0− 3 =−1−3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2 =
13
m( LM ) = −3 + 2
0− 3 =−1−3
=13
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2 =
13
m( LM ) = −3 + 2
0− 3 =−1−3
=13
m( KL ) = −2 − 4
3 − 1
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2 =
13
m( LM ) = −3 + 2
0− 3 =−1−3
=13
m( KL ) = −2 − 4
3 − 1 =−62
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2 =
13
m( LM ) = −3 + 2
0− 3 =−1−3
=13
m( KL ) = −2 − 4
3 − 1 =−62 = −3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2 =
13
m( LM ) = −3 + 2
0− 3 =−1−3
=13
m( KL ) = −2 − 4
3 − 1 =−62
m( JM ) = −3 − 3
0+ 2
= −3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2 =
13
m( LM ) = −3 + 2
0− 3 =−1−3
=13
m( KL ) = −2 − 4
3 − 1 =−62
m( JM ) = −3 − 3
0+ 2 =−62
= −3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
EXAMPLE 4Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52
= 25 + 25 = 50
KM = (1− 0)2 + (4 + 3)2 = 12 + 72
= 1+ 49 = 50
m( JK ) = 4 − 3
1+ 2 =
13
m( LM ) = −3 + 2
0− 3 =−1−3
=13
m( KL ) = −2 − 4
3 − 1 =−62
m( JM ) = −3 − 3
0+ 2 =−62 = −3
= −3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Monday, April 22, 2013
PROBLEM SET
Monday, April 22, 2013
PROBLEM SET
p. 422 #1-31 odd, 41, 49, 55, 59, 61
“Character - the willingness to accept responsibility for one's own life - is the source from which self respect
springs.” - Joan DidionMonday, April 22, 2013