Activity
Lab
Activity Lab
Transforming to Find Area
Transforming
to Find Area
FOR USE WITH LESSONS 10-1 AND 10-2
You can use transformations to find area formulas.
Students draw and cut polygons
into pieces to form different
polygons to develop the area
formulas for parallelograms,
triangles, and trapezoids.
In these activities, you will cut polygons into pieces. You will use isometry
transformations on the pieces to form different polygons. Because a preimage
and its image are congruent and congruent figures have the same area
(Postulate 1-9), you can find area formulas for the new polygons.
You will need several pieces of grid paper. Let the side of each grid square
represent one unit of length.
Guided Instruction
English Language Learners ELL
1
Review the term transformation
as a change in position, size, or
shape of a geometric figure.
Ask: What are some types of
transformations that you have
studied? rotations, reflections,
translations, glide reflections
• For the parallelogram shown here, count
and record the number of units in the base
of the parallelogram. Do the same for the
height.
• Copy the parallelogram onto grid paper.
• Cut out the parallelogram. Then cut it into
two pieces as shown.
• Translate the triangle to the right through a
distance equal to the base of the
parallelogram.
Activity 1
Teaching Tip
The translation image is a rectangle. The
parallelogram and the rectangle have the same
area.
Have students repeat the activity
using different sized and shaped
parallelograms. Students will
observe that a rectangle is
always formed.
Special Needs
1. For the rectangle, how many units are in the base? The height? 9; 5
2. How do the base and the height of the rectangle compare to the base and
height of the parallelogram? They are the same.
L1
Before students try answering
Exercises 1-3, ask: Why must the
rectangle and parallelogram have
the same area? The parallelogram
and rectangle are formed from
the same pieces.
ACTIVITY
3. Write the formula for the area of the rectangle. Explain why you can use this
formula to find the area of a parallelogram. See margin.
2
ACTIVITY
Now you have the formula for the area of a
parallelogram. You can use transformations and this
formula to find an area formula for a triangle.
Teaching Tip
• Count and record the base and height of the
triangle.
• Copy the triangle. Mark the midpoints A and B
and draw the midsegment AB.
• Cut out the triangle. Then cut it along AB.
• Rotate the small triangle 180° about point B.
Ask: What are three ways
rectangles and parallelograms are
the same? equal base lengths,
heights, and areas
Resources
The bottom part of the original triangle and the image
of the top part form a parallelogram.
Students use scissors and
rectangular dot paper or graph
paper.
4. For the parallelogram, how many units are in the
base? The height? 8; 3
532
Activity Lab Transforming to Find Area
3. A ≠ bh; explanations may
vary. Sample: A
parallelogram can be
transformed to a rectangle
with the same area, base,
and height.
532
A
A
B
B
5. How do the base and height of the parallelogram compare to the base and
height of the original triangle? Write an expression for the height of the
parallelogram in terms of the height, h, of the triangle.
The bases are the same; the height of the parallelogram is half the height of the triangle; 1
2 h.
6. Write your formula for the area of a parallelogram from Exercise 3. To find an
area formula for a triangle, substitute into the formula the expression you
wrote in Exercise 5 for the height of the parallelogram.
A = bh; A = 1
2bh
3
ACTIVITY
You can find an area formula for a trapezoid using a transformation similar to the
one you used for a triangle.
• Draw a trapezoid like the one shown here.
• Count and record its bases and height.
• Find the midpoints of the legs and the
midsegment of the trapezoid.
• Cut out the trapezoid. Then cut it along the
midsegment.
Have each student draw a
different parallelogram and cut it
out. Each student then cuts their
parallelogram along a diagonal,
forming two triangles. Ask: How
does the area of each triangle
compare to the area of the
parallelogram? The area of each
triangle is half the area of the
parallelogram.
Special Needs
8. a. Write an expression for the base of the parallelogram in terms of the two
bases, b1 and b2, of the trapezoid. b1 + b2
b. Write an expression for the height of the parallelogram in terms of h, the
height of the trapezoid. 1
2h
c. To find an area formula for a trapezoid, substitute your answers for parts (a)
and (b) into your area formula for a parallelogram. A = 1
2 (b1 + b2)h
Teaching Tip
Have students repeat the activity
for different sized and shaped
trapezoids. Students will observe
that a parallelogram is always
formed. Students who draw
trapezoids with two right angles
will observe that their two
trapezoids form a rectangle.
EXERCISES
9. In Activity 2, is there another rotation of the small triangle that will form a
parallelogram? Explain. yes; rotate 180° about point A.
10. Make another copy of the Activity 2 triangle. Mark the midpoints A and B.
Find a rotation of the entire triangle so that the preimage and image together
form a parallelogram. Show how to use it and your formula for the area of a
parallelogram to find the formula for the area of a triangle. See margin.
Exercises
Have students work with partners
or in small groups to complete
the exercises.
11. For Exercise 10, there are in fact three rotations of the entire triangle that you
can use. Find all three and describe them. See margin.
12. Here is the trapezoid with a different cut. What
transformation can you apply to the top piece to form
a triangle from the trapezoid? Use your formula for
the area of a triangle to find a formula for the area
of a trapezoid. Rotate the top piece 180° about
point N; A = 1
2 (b1 + b2)h
L1
After students arrange the two
trapezoids, ask: What do you
notice about the horizontal
sides of your figure? They are
congruent and parallel.
What type of figure do the two
trapezoids form? parallelogram
7. What transformation can you apply to the top piece of the trapezoid to form a
parallelogram? Rotate 180° about point N.
N
13. Show how you can find an area formula for a kite using a
reflection. (Hint: Reflect half of the kite across its line of
symmetry d1 by folding the kite along d1. How is the area
of the triangle formed related to the area of the kite?)
See margin.
d1
d2
Activity Lab Transforming to Find Area
10. Rotate the entire k 180°
about A (or B) to form a $
that has the same base b
and same height h as the
k and is twice the size of
the triangle. Then bh ≠
area ($) ≠ 2(area (k)), so
area (k) ≠ 12 bh.
Alternate Method
Activity 3
N
M
Activity 2
11. Rotate the entire k 180°
about the midpoint of any
of its three sides.
13. The area of each k is half
the area of the kite. Area
(kite) ≠ 2(area (k)) ≠
2 Q 12 bh R ≠ bh where b ≠ d1
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and h ≠ 12d2, so the area
(kite) ≠ 12d1d2 (half the
product of its diagonals).
533