Download Pythagoras and President Garfield

Activity 2: Pythagoras and President Garfield (Grades 6-10)
Objectives:



Understand the statement of the Pythagorean theorem
Apply the Pythagorean theorem to solve for unknown sides of right triangles
Prove the Pythagorean theorem using the method developed by President
James Garfield
NCTM Standards:
Standard 3: Geometry and Spatial Sense: analyze characteristics and properties of
two-dimensional objects.
Standard 7: Reasoning and Proof: develop and evaluate mathematical arguments
and proofs.
The Pythagorean Theorem
The Pythagorean theorem is one of the most well known and important theorems in
all of mathematics. It is very useful for finding a side of a right triangle when only
two are known. It is also useful to help one establish whether or not a triangle is a
right triangle (if it has a 90-degree angle).
The Pythagorean theorem states that a triangle is a right triangle if and only if the
square of the longest side equals the sum of the squares of the two other sides. In
other words, the triangle in Figure 1 is a right triangle if and only if x 2+ y2= z2.
Figure 1. A triangle.
1. Use the Pythagorean theorem to find the unknown side of each triangle.
2. A builder is framing a house. She wants to make sure the walls are at right
angles to each other. She makes the measurements of two walls and the diagonal,
as shown in Figure 2. Use the Pythagorean theorem to help the builder decide if the
walls are at right angles.
Figure 2. Builder's diagram.
The President's Proof
There are many proofs to the Pythagorean theorem. President James Garfield
developed his own proof in The Journal of Education (Volume 3 issue161) in 1876.
President Garfield studied math at Williams College (in Williamstown, MA) and
taught in the public school in Pownal, Vermont, for a year or two after graduating.
President Garfield may have been joking when he stated about his proof that, "we
think it something on which the members of both houses can unite without
distinction of the party." A nice feature of mathematical proofs is that they are not
subject to political opinion.
See if you can follow the president's proof.
See Figure 3. Starting with the right triangle ABC, construct a line perpendicular to
BC and through the point C. Extend AC from the point C.
Figure 3. Diagram of triangle ABC for use in the president's proof.
Now look at Figure 4. Draw a circle with center C and radius equal to the length of
BC. Label the point E as the point where the circle and the perpendicular line
intersect. Draw a line through E and perpendicular to AC and label the intersection
point D. Lastly, draw the segment BE.
Figure 4. Diagram for use with the president's proof.
3. What can you say about triangles ABC and CDE?
4. What is the area of
a. triangle ABC?
b. triangle BCE?
c. triangle CDE?
5. What type of quadrilateral is ABDE?
6. What is the area of ABDE?
7. What can be said about the sum of the area of triangles ABC, BCE, and CDE and
the area of ABDE?
8. Using only x's, y's and z's, write an equation for the sum of the area of triangles
ABC, BCE, and CDE and for the area of ABDE.
9. Solve the equation you found in problem 8.
10. What have you proven?
Student Activity Page
April 2000
Activity 2: Pythagoras and President Garfield
A. The Pythagorean Theorem
The Pythagorean theorem is one of the most well known and important theorems in all of
mathematics. It is very useful for finding a side of a right triangle when only two are
known. It is also useful to help one establish whether or not a triangle is a right triangle
(if it has a 90-degree angle).
The Pythagorean theorem states that a triangle is a right triangle if and only if the square
of the longest side equals the sum of the squares of the two other sides. In other words,
the triangle in Figure 1 is a right triangle if and only if x2+ y 2= z2.
B
z
x
A
y
C
Figure 1. A triangle.
1. Use the Pythagorean theorem to find the unknown side of each triangle.
a.
b.
5
8
13
9
2. A builder is framing a house. She wants to make sure the walls are at right angles to
each other. She makes the measurements of two walls and the diagonal, as shown in
Figure 2. Use the Pythagorean theorem to help the builder decide if the walls are at right
angles.
20
12.5
15
Figure 2. Builder’s diagram.
B. The President’s Proof
There are many proofs to the Pythagorean theorem. President James Garfield developed
his own proof in The Journal of Education (Volume 3 issue161) in 1876. President
Garfield studied math at Williams College (in Williamstown, MA) and taught in the
public school in Pownal, Vermont, for a year or two after graduating. President Garfield
may have been joking when he stated about his proof that, “we think it something on
which the members of both houses can unite without distinction of the party.” A nice
feature of mathematical proofs is that they are not subject to political opinion.
See if you can follow the president’s proof.
See Figure 3. Starting with the right triangle ABC, construct a line perpendicular to BC
and through the point C. Extend AC from the point C.
B
z
x
A
y
C
Figure 3. Diagram of triangle ABC for use in the president’s proof.
Now look at Figure 4. Draw a circle with center C and radius equal to the length of BC.
Label the point E as the point where the circle and the perpendicular line intersect. Draw
a line through E and perpendicular to AC and label the intersection point D. Lastly, draw
the segment BE.
E
B
z
x
A
y
C
D
Figure 4. Diagram for use with the president’s proof.
3. What can you say about triangles ABC and CDE?
4. What is the area of
a. triangle ABC?
b. triangle BCE?
c. triangle CDE?
5. What type of quadrilateral is ABDE?
6. What is the area of ABDE?
7. What can be said about the sum of the area of triangles ABC, BCE, and CDE and the
area of ABDE?
8. Using only x’s, y’s and z’s, write an equation for the sum of the area of triangles ABC,
BCE, and CDE and for the area of ABDE.
9. Solve the equation you found in problem 8.
10. What have you proven?
April 2000
Activity 2: Pythagoras and President Garfield
Solutions
A. The Pythagorean Theorem
8 2 + 9 2 ≈ 12
1. a.
b.
13 2 − 5 2 = 12
2. Since 12.52 + 152 = 381.25 and 202 = 40,0 the corners are not square. The angle between the two
shorter sides needs to be reduced until the long side is approximately 19.5.
B. The President’s Proof
3. They are congruent. By construction, side BC and CE are the same. Angle ECD is the same as angle
ABC because exterior angle ECD + the Right Angle = Angle ABC + Right Angle interior to triangle
ABC. Since the hypotenuse and an angle are congruent, the two triangles are congruent. Now they have
CD = x and DE = y
4. a. ½ x y
b. ½ z z = ½ z2
c. ½ xy
5. Trapezoid
6. Area of a trapezoid is 1/2 the height times the average of the bases so,
½ (x+y)(x+y) = ½ (x2 + 2xy + y2) = ½ x2 + xy + ½ y2
7. They are equal since they occupy the same space.
8. ½ x y + ½ z2 + ½ x y = ½ x2 + xy + ½ y2
9. xy + ½ z2 + = ½ x2 + xy + ½ y2
½ z2 = ½ x2 + ½ y2
z2 = x2 + y2
10. Students have proven the Pythagorean theorem.