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Lesson 9.1 • The Theorem of Pythagoras
The Pythagorean Theorem states a relationship between the areas
of squares constructed on the three sides of a right triangle. In this
investigation you’ll explore this relationship in two ways: first, by
measuring and calculating and, second, by cutting the squares on the
two legs of the right triangle and arranging them to fit in the square
on the hypotenuse.
Investigation: The Three Sides of a Right Triangle
Sketch
Step 1
!.
In a new sketch, construct AB
Step 2
!.
Construct the midpoint C of AB
Step 3
Construct a circle centered at C with radius endpoint B.
Step 4
!! and BD
!, where point D is a point on the circle.
Construct AD
Step 5
Hide the circle and the midpoint. Drag points A, B, and D to be
sure your triangle stays a right triangle.
Step 6
Open a sketch that has a Custom tool for creating a square. This
can be either a sketch you have made yourself or the sketch
Polygons.gsp.
Step 7
Construct squares on the outsides of the triangle.
Step 8
Measure the areas of the three squares.
D
D
A
C
B
B
A
Steps 1–4
Step 7
Investigate
1. Drag vertices of the triangle and look for a relationship among
the areas of the three squares. Use the calculator to confirm
your observation. State a conjecture about these three areas
(The Pythagorean Theorem).
(continued)
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CHAPTER 9
Discovering Geometry with The Geometer’s Sketchpad
©2008 Key Curriculum Press
DG4GSP_897_09.qxd
12/20/06
1:33 PM
Page 123
Lesson 9.1 • The Theorem of Pythagoras (continued)
Sketch
You can use dissection puzzles to demonstrate that figures have equal areas
without measuring them. Continue in your sketch to create a dissection
that demonstrates the Pythagorean Theorem.
Step 9
Delete the square interiors on the longer leg and on the hypotenuse,
but not on the smaller leg.
Step 10
Find point O, the center of the square on the longer leg, by
constructing the diagonals.
Step 11
Hide the diagonals.
Step 12
Construct a line through point O parallel to hypotenuse AB.
Step 13
Construct a line through point O perpendicular to hypotenuse AB.
Step 14
Construct the four points where these two new lines intersect the
sides of the square.
Step 15
Construct the polygon interiors of the four quadrilaterals in the
larger square. Give them all different colors.
Step 16
Hide the lines.
D
O
O
D
B
A
B
A
Steps 9 and 10
Steps 12–16
Investigate
2. You should now have a five-piece puzzle. One piece is the square on
the small leg, and four pieces form the square on the longer leg. The
object of the puzzle is to arrange these five pieces to fit in the square
on the hypotenuse. In turn, select each of the five interiors, choose Cut
from the Edit menu, then choose Paste. Hide any extraneous points.
Your five pieces are now free to be moved. Click in a blank area of
your sketch to deselect everything, then drag each piece and find a way
to fit all the pieces in the square on the hypotenuse. When you finish,
draw the solution on your paper. (That is, draw a square and sketch
how the five pieces fit into the square.)
3. Explain how this dissection demonstrates the Pythagorean Theorem.
Discovering Geometry with The Geometer’s Sketchpad
©2008 Key Curriculum Press
CHAPTER 9
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