Advanced Math Topics: Proof of Distance Formula

Week of Monday, March 10-Friday, March 14, 2008

Advanced Math Topics

Objectives

• SWBAT use formal algebraic deductions to prove simple geometric area formulas.
• SWBAT use deductive rules to craft proofs of algebraic formulas.

PA Standards

• PA 2.4.5.E. Interpret statements made with precise language of logic (e.g., "all", "or", "every", "none", "some", "or", "many").
• PA 2.4.8.C. Use if...then statements to construct simple, valid arguments.
• PA 2.4.11.A. Use direct proofs, indirect proofs or proof by contradiction to validate conjectures.
• PA 2.4.11.B. Construct valid arguments from stated facts.
• PA 2.4.11.C. Determine the validity of an argument.
• PA 2.9.5.A. Give formal definitions of geometric figures.

Statement of the Problem

I claim:

Given two points in a plane and , the distance between them may be calculated as .

I further claim that we may determine this starting from the area of a rectangle:

The area of a rectangle with height and width may be found with .

The Plan Of Attack

1. Use the rectangle area formula to derive a formula for the area of a square.
   1. Be rigorous!
   2. Define a square: A square is a rectangle with all four sides equal.
   3. So a square is a rectangle.
   4. So if the square has sides of length , the area of that rectangle with sides and is .
2. ... and, from that, a rule from the area of a square to the length of each side.
   1. If we have the area of a square, we can solve to .
3. Use the rectangle area formula for the area of a right triangle.
   1. Draw a right triangle with height and width .
   2. Duplicate it so that there is a rectangle with height and width .
   3. Since there are two triangles, and they must sum to the area of the full rectangle , the area of each must be .
4. Use the above derivations to prove the Pythagorean Theorem:

   Within every right triangle whose legs are of length and and whose hypotenuse is of length , the lengths of the sides will have the relationship .

   1. Graded Classwork: Using the Internet or any other resource, find a proof of the Pythagorean Theorem which you can understand and explain. Write it in your own words and print it out or email it to me at [email protected]
   2. Sketch a right triangle, label its sides , , and , and duplicate it four times, as in this illustration.
   3. Calculate the area of each right triangle , then multiply by four to determine the area of all four right triangles together .
   4. Calculate the area of the large square .
   5. Subtract the area of the four triangles, leaving only the area of the inner square .
   6. Since the inner square has sides of length , its area is also and, by transitivity, .
5. Define the distance between two points and on the same axis (number line) as . So our only real problem is to prove the distance between two points which are diagonal from each other.
   1. This is just a definition. Explain it. Notice the absolute-value bars; the distance between two points on a number line is always (positive and) the same whichever way you look at the points because distance does not have direction.
   2. (If they're curious, mention vectors, which have both distance and direction.)
6. Construct a right triangle using our diagonal points and , and a third point .
   1. Note that the orientation of this triangle may be different from in this diagram. Is that a problem? No.
7. Use the smaller distance formula (step 5) to determine the lengths of both legs of this right triangle.
   1. The horizontal leg will have length .
   2. The vertical leg will have length .
8. Use the Pythagorean theorem to determine the length of the hypotenuse of this right triangle... which is the distance between our original two points.
   1. Since this is a right triangle, the hypotenuse will have length .
   2. Since we're squaring those inner terms, the absolute-value bars don't matter any more: .
9. Demonstrate that the smaller distance formula works with this new distance formula, so we need only use regardless of where on the plane the points are.

Assessments

Proof-related questions:

1. Given a parallelogram with measurements as shown, can you prove that its area is still ?
2. Given a triangle with measurements as shown, can you prove that its area is still ?
3. Given a kite with diagonals and , can you prove that its area is ?