G.GMD.1 Wkst 4

G.GMD.1 STUDENT NOTES & PRACTICE WS #4
1
THE TRAPEZOID
The trapezoid has the most unique formula of all the basic polygons and very few students know where it
comes from. It is for this reason that many students struggle to remember it. For them it has no meaning and
no context. Unlike a triangle being half of a rectangle or a parallelogram being a sheared rectangle, the
trapezoid is just a formula. Where does the Area = ½ (b1 + b2)h come from?
Before demonstrating this we need to first discuss some particular terms to this polygon. A trapezoid has two
parallel sides; these two sides are called the bases of the trapezoid. They are usually denoted as b1 and b2 for
base 1 and base 2. If the trapezoid is an isosceles trapezoid then the two non-base sides are congruent and
are called the legs of the trapezoid, just as we do in an isosceles triangle.
b1
b1
h
b1
leg
h
b2
leg
h
b2
b2
TRAPEZOID AREA – HALF OF THE WHOLE (DOUBLING TECHNIQUE)
One way to work with a trapezoid is to create two of them, thus doubling the area. To create the second
trapezoid you rotate the original about the midpoint of one of the non-base sides. In doing this you will create
either a rectangle or a parallelogram with the same height and a base of (b1 + b2). The area of a
rectangle/parallelogram is (b1 + b2)(height), thus making the area of the trapezoid ½ (b1 + b2)(height).
b1
b1
h
h
h
b2
b2
b1
b1
h
h
b2
b1
b 1 + b2
h
b2
b1
b1 + b2
AREATRAPEZOID = 1 (b1 + b2)h
2
TRAPEZOID AREA – DISSECTION (HALVING TECHNIQUE)
We can also use dissection to demonstrate this wonderful formula. Doing a similar technique as we did with
the triangle we can cut the height in half to create some pieces and then rearrange them to form a rectangle.
G.GMD.1 STUDENT NOTES & PRACTICE WS #4
2
b1
b1
b1
h
h
h
1
b2
b2
b2
b1
b1
b1
h
h
h
b1
2
1
2
b2
b2
h
b1 + b2
b2
b1
h
b1 + b2
This creates a rectangle with dimensions, ½ the height of the trapezoid and the base of length b1 + b2.
TRAPEZOID – DISSECTION (THE PIECES MAKE THE WHOLE)
A trapezoid is really just a composite shape – a shape made up of two or more other shapes and depending on
which shapes you see within the trapezoid will depend on how you come about deriving the formula for the
trapezoid. This is a wonderful exercise for a class – to try to derive the formula for a trapezoid based on the
different composite shapes within the trapezoid.
b1
b1
b1
h
h
b2
b1
h
b2
h
b2
b2
Let’s do two examples to show you how these work.
A trapezoid is a rectangle and a triangle.
The key to success in this method is the correct
naming of b2 when it is divided into two pieces, the
rectangle makes the opposite side congruent b1 and
then what would be left of b2 would be b2 – b1.
b1
h
b2
Arearectangle + Areatriangle = Areatrapezoid
h
h
b1
1
(b2 − b1 ) h = Areatrapezoid
2
1
1
b1h + b2 h − b1h = Areatrapezoid
2
2
1
1
b1h + b2 h = Areatrapezoid
2
2
1
h (b1 + b2 ) = Areatrapezoid
2
b1h +
b1
b2 - b1
G.GMD.1 STUDENT NOTES & PRACTICE WS #4
3
A trapezoid is two triangles.
The key to success in this method is the see that both
triangles have the height of h. The triangle with b1 as
its base has a height of h as well.
b1
h
Areatriangle + Areatriangle = Areatrapezoid
b2
1
1
b1h + b2 h = Areatrapezoid
2
2
1
h (b1 + b2 ) = Areatrapezoid
2
b1
h
b2
There are lots of ways to divide a trapezoid up differently to yield quite easy level algebra operations by they
all simplify to the same conclusion that A = ½ (b1 + b2)h.
Given the following trapezoids, calculate their areas.
5 cm
8 cm
8 cm
4 cm
13 cm
5 cm
14 cm
22 cm
Area = _____________
Area = _____________
A = ½ (5 + 22)(4)
A = 54 cm2
A = ½ (8 + 22)(8)
A = 120 cm2
3 cm
Area = _____________
52 + x2 = 132
x = 12
A = ½ (8 + 20)(5)
A = 70 cm2
6 cm
3 cm
4 cm
8 cm
11 cm
A = ½ (3 + 11)(4)
A = 56 cm2
60°
4 cm
h = 4 3 cm
h2 + 12 = 32
h = 2 2 cm
b1 = 4 3 cm b2 = 4 + 4 3 cm
A = ½ ( 4 3 + 4 + 4 3 )( 4 3 )
A = ½ (6 + 8)( 2 2 )
A = (7)( 2 2 )
A = ½ (4 + 8 3 )( 4 3 )
A = ½ (16 3 + 96)
A = 8 3 + 48 cm2
A = 14 2 cm2
G.GMD.1 WORKSHEET #4
NAME: ____________________________ Period _______
1. What is the definition of a trapezoid?
2. A trapezoid can be thought of as a composite shape (a shape made up of smaller more basic shapes).
Dissect the trapezoid into the required shapes.
a) Two Triangles
b) Rectangle & Triangle
c) Parallelogram and
Triangle
d) Two Triangles and a
Rectangle
3. One way to demonstrate that the area formula for a trapezoid is A = ½ (b1 + b2)h is using a DOUBLING
TECHNIQUE. Demonstrate how rotating a trapezoid on the midpoint of one of its non-base sides helps us
discover its formula for area. (Use patty paper to help you)
b1
h
b2
4. Another technique is the HALVING TECHNIQUE. Use dissection to cut the height of the trapezoid in half
(midpoints have been provided). How can this help us derive the formula for the area of a trapezoid? (Use
patty paper to help you.)
b1
h
b2
1
G.GMD.1 WORKSHEET #4
2
5. Use dissection to establish the area formula for a trapezoid. Create the relationship based on the
dissection and then transform that into the area formula for a trapezoid.
a) A Rectangle and a Triangle
b) A Parallelogram and a Triangle
b1
b1
h
h
b2
b2
c) Two Triangles
d) A Rectangle and Two Triangles
b1
b1
h
h
x
b2
b2
6. If a trapezoid is an isosceles trapezoid, it has two congruent legs and two sets of congruent base angles.
Use dissection to determine the area formula.
b1
h
b2
G.GMD.1 WORKSHEET #4
3
7. Determine the area of the following trapezoids.
a)
b)
c)
6 cm
3 cm
45°
4 2 cm
5 cm
7 cm
5 cm
11 cm
8 cm
Area = ____________
d)
Area = ____________
e)
6 cm
Area = ____________
f)
10 cm
5 cm
6 cm
5 cm
12 cm
60°
8 cm
Area = ____________
g)
10 cm
Area = ____________
h)
6 cm
Area = ___________________ (E)
i)
5 cm
13 cm
10 cm
10 cm
6 cm
60°
14 cm
30°
14 cm
13 cm
12 cm
Area = ____________
j)
60°
Area = ____________
k)
60°
2 cm
Area = ____________
l)
2 3 cm
8 cm
8 cm
6 cm
30°
12 cm
Area = _________________ (E)
30°
10 cm
Area = ____________
Area = ____________
G,GMD.I
WORKSHEET #4
1. What is the definition of a trapezoid?
k
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.
2. A trapezoid can be thought of as a composite shape (a shape made up of smalter more basic shapes).
Dissect the trapezoid into the required shapes.
a)Two Triangles
b) Rectangle &
Triangle
+A
c) Parallelogram
Triangle
0
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(
\
zt( rrr.nn- ta"r^.
1 a**u('
and
d) Two Triangles and a
Rectangle
3. One way to demonstrate that the area formula for a trapezoid is A=Tz (br + bzlh is using a DOUBLING
TECHNIQUE. Demonstrate how rotating a trapezoid on the midpoint of one of its non-base sides helps us
discover its formula for area. (Use patty paper to help you)
4. Another technique is the HALVING TECHNTQUE. Use dissection to cut the height of the trapezoid in hatf
(midpoints have been provided). How can this help us derive the formula for the area of a trapezoid? (Use
patty paper to help you.)
br
A Va.rolv\aqvo<,-= Q r)(u, + b)
lr4l-
J
A-yn
4"ro;a.
- &Q (a,*l')
G.GMD,I WORKSHEET#4
5. Use dissection to establish the area formula for a trapezoid. Create the relationship based on the
dissection and then transform that into the area formula for a trapezoid.
a)A Rectangle and a Triangle
^W
b)A Parallelogram and a Triangle
'br
lb2
br
,[-\
"_L
'-. \
bz
h (ur-u,\
hIr,\ +
c)Two Triangles
d)A Rectangle and Two Triangles
br
b1
b2
+
ft= )b,t' + i b,r
ta,
-L
2 k$,
5. lf a trapezoid is an isosceles trapezoid, it has two bongruent legs and two sets of congruent base angles.
Use dissection to determine the area fbrmula.
-I
F
G.GMD.I WORKSHEET #4
3
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i
7. Determine the area of the following trapezoids.
a)
b)
c)
A,l'ffi,,
3cm
11 cm
8cm
)P+.ilk)
Area =
d)
Lu)kri
38., cmu
Area =
Area
f)
e)
=
23
CynL
"/^N'.:i7i{::'
t'-b
I L_J"'
12 cm
8cm
10 cm
I> (rl (rt *)
bi 0\,
(E)
y2+n'.17L
NL
l-y4y=lb?
?t--zl
v--{
14 cm
13 cm
I (r) ( u+w)
12 cm
rr6a (tv
Area =
5o ca-
n)
l3F cwrz
i)
zrE.,
12 cm
thq\
10 cm
(e+zo)
Area= 6LlG
crrl
f'E
(E)
Area =
Area
= t{{,
clu.L
1(:) (rc
+'6)