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