**APPLICATIONS**

Solve the following problems. In solving these problems, keep in mind that

it is always helpful to draw pictures to guide your intuition and reasoning.

**
RAILROAD LINE PROBLEM:** A high-speed railroad line was built through

the desert. In order to reduce derailments along a two-mile stretch, the

track was made using straight rails one mile long. The rails were laid in the

winter, and in the summer, due to the heat, each rail expanded one foot in

length. Ordinarily, the rails would push each other to the side, but in this

case they jutted upwards where the ends meet. How high above the

ground are the rails at the joints? (Historical note: when the RT Metro in

Sacramento, California was first running, the trains experienced a problem

similar to this situation.)

(This problem is adapted from College Preparatory Mathematics, Mathematics 1 (Algebra 1, Units 7-2) v. 3.2)

NOT DRAWN TO SCALE |

**EUCLID STREET PROBLEM: **In order to go to school
each day, Michelle

walks about 3 miles east from her home on Pythagoras Avenue, then walks

north along Euclid Street to the front of her school. She knows that the

shortest distance from her home to the front of the school is about 9 miles

(as the crow flies). How far does she walk along Euclid Street each school

day?

**RECTANGULAR BOX PROBLEM:** You have a huge
rectangular box with

dimensions 9 ft. x 12 ft. x 15 ft. What is the length of the longest rod you

could fit in your box?

**CYLINDRICAL CAN PROBLEM:** The circumference of the
base of a right

cylindrical can is 24 inches; the height is 7 inches. A shortest possible

spiral that winds once from the top to the bottom (see below) is painted on

the can, so that the bottom of the spiral is directly below the top of the

spiral. What is the length of the spiral?

**JOURNAL**

Four proofs of the Pythagorean theorem are sketched in the following

pages. Select one of the four proofs, study it, and figure out why it works.

Use the fourfold way to explain the proof. Be prepared to share your

findings with others.

**FOUR PROOFS
First Proof**

The Pythagorean theorem is well known to most high school geometry

students. What is not so well known is that there are numerous ways to

prove this famous theorem. On this page you have a diagram that

suggests a particular version of the proof. Your task is to complete the

proof along the lines suggested below. Be ready to present your proof to

the class.

Proof sketch: Let A be the area of the entire figure. Let
T be the area of the

right triangle with legs a and b and hypotenuse c. The figure contains four

congruent copies of the right triangle. (Why are they congruent?)

Decompose A in two different ways to see that

A = a^{2} + b^{2} + 2T,

A = c^{2} + 2T.

Conclude that a^{2} + b^{2} = c^{2}.

Second Proof

James Abram Garfield (1831-1881) was elected the country’s twentieth

president in 1880. Earlier he had taught mathematics, and he discovered a

proof of the Pythagorean theorem in 1876, while he was a member of the

House of Representatives. Find Garfield’s proof of the Pythagorean

theorem using the diagram below and expressing the area of the trapezoid

in two different ways. Be ready to present your proof to the class.

Proof sketch: Let A be the area of the entire figure,
which is a trapezoid.

The area formula for the trapezoid is

(Why is this formula true? You can find this formula by
joining two copies

of the trapezoid together along the slanted edge to form a rectangle.)

Decompose the trapezoid into three triangles and express A as the sum of

the areas of three triangles,

Equate these two expressions for A and conclude that

c^{2} = a^{2} + b^{2}.

**Third Proof**

On this page you have a diagram that suggests yet another version of the

proof. Your task is to prove the theorem using the decomposition of the

larger square suggested by the diagram below. Be ready to present your

proof to the class.

Proof sketch: Assume a ≤ b . Let A be the area of the
largest square. Then

A = (a+b)^{2}.

Note that the side-length of the inside square is b-a. (Why?) Decompose A

into four rectangles and the inside square, to see that

A = 4ab + (b-a)^{2} .

Equate these two expressions for A and conclude that

c^{2} = a^{2} + b^{2}.

Fourth Proof

On this page you have a diagram that suggests a version of the proof that

is closely related to the third proof. (How is it related?) Your task is to

prove the theorem using the decomposition of the square suggested by

the diagram. Be ready to present your proof to the class.

Proof sketch: Assume a ≤ b. Let A denote the area of the
outside square.

Then

A = c^{2}.

The inside figure is a square (why?) with sides of length a - b (why?).

Decompose A into four triangles and the inside square, to see that

Equate these two expressions for A and conclude that

c^{2} = a^{2} + b^{2}.

**POINT A TO POINT B**