### Rectangles

```• rectangle—a parallelogram with right angles
Use Properties of Rectangles
CONSTRUCTION A rectangular garden gate is
reinforced with diagonal braces to prevent it from
sagging. If JK = 12 feet, and LN = 6.5 feet, find KM.
Use Properties of Rectangles
Since JKLM is a rectangle, it is a parallelogram. The
diagonals of a parallelogram bisect each other, so
LN = JN.
JN + LN = JL
LN + LN = JL
Substitution
2LN = JL
2(6.5) = JL
13 = JL
Simplify.
Substitution
Simplify.
Use Properties of Rectangles
JL  KM
If a is a rectangle,
diagonals are .
JL = KM
Definition of congruence
13 = KM
Substitution
Quadrilateral EFGH is a rectangle. If GH = 6 feet
and FH = 15 feet, find GJ.
A. 3 feet
B. 7.5 feet
C. 9 feet
D. 12 feet
A.
B.
C.
D.
A
B
C
D
Use Properties of Rectangles and Algebra
Quadrilateral RSTU is a rectangle. If mRTU =
8x + 4 and mSUR = 3x – 2, find x.
Use Properties of Rectangles and Algebra
Since RSTU is a rectangle, it has four right angles.
So, mTUR = 90. The diagonals of a rectangle bisect
each other and are congruent, so PT  PU. Since
triangle PTU is isosceles, the base angles are
congruent so RTU  SUT and mRTU = mSUT.
mSUT + mSUR = 90
mRTU + mSUR = 90
Substitution
8x + 4 + 3x – 2 = 90
Substitution
11x + 2 = 90
Use Properties of Rectangles and Algebra
11x = 88
x = 8
Subtract 2 from each
side.
Divide each side by 11.
Quadrilateral EFGH is a rectangle. If mFGE =
6x – 5 and mHFE = 4x – 5, find x.
A. x = 1
B. x = 3
C. x = 5
D. x = 10
A.
B.
C.
D.
A
B
C
D
Proving Rectangle Relationships
ART Some 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 and the diagonals of the
frame. If AB = 12 inches,
BC = 35 inches, CD = 12 inches,
DA = 35 inches, BD = 37 inches,
and AC = 37 inches, explain how
an artist can be sure that the
frame is rectangular.
Proving Rectangle Relationships
Since AB = CD, DA = BC, and AC = BD, AB  CD,
DA  BC, and AC  BD.
Answer: Because AB  CD and DA  BC, ABCD is a
parallelogram. Since AC and BD are
congruent diagonals in parallelogram ABCD,
it is a rectangle.
Max is building a swimming pool in his
backyard. He measures the length and width
of the pool so that opposite sides are
parallel. He also measures the diagonals of
the pool to make sure that they are
congruent. How does he know that the
measure of each corner is 90?
A. Since opp. sides are ||, STUR
must be a rectangle.
B.
Since opp. sides are , STUR
must be a rectangle.
C.
Since diagonals of the are ,
STUR must be a rectangle.
D.
STUR is not a rectangle.
A.
B.
C.
D.
A
B
C
D
Rectangles and Coordinate Geometry
Quadrilateral JKLM has vertices J(–2, 3), K(1, 4),
L(3, –2), and M(0, –3). Determine whether JKLM is
a rectangle using the Distance Formula.
Step 1
Use the Distance
Formula to determine
whether JKLM is a
parallelogram by
determining if opposite
sides are congruent.
Rectangles and Coordinate Geometry
Since opposite sides of a quadrilateral have the same
measure, they are congruent. So, quadrilateral JKLM
is a parallelogram.
Rectangles and Coordinate Geometry
Step 2
Determine whether the diagonals of
are congruent.
JKLM
Answer: Since the diagonals have the same
measure, they are congruent. So JKLM is
a rectangle.
Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3),
Y(3, 1), and Z(2, –1). Determine whether WXYZ is a
rectangle by using the Distance Formula.
A. yes
B. no
C. cannot be
determined
1.
2.
3.
A
B
C
Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3),
Y(3, 1), and Z(2, –1). What are the lengths of
diagonals WY and XZ?
A.
B. 4
C. 5
D. 25
A.
B.
C.
D.
A
B
C
D
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