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Private: Learning Math: Geometry
The Pythagorean Theorem
Continue to examine the idea of mathematical proof. Look at several geometric or algebraic proofs of one of the most famous theorems in mathematics: the Pythagorean theorem. Explore different applications of the Pythagorean theorem, such as the distance formula.
In This Session
Part A: The Pythagorean Theorem
Part B: Proving the Pythagorean Theorem
Part C: Applications of the Pythagorean Theorem
Homework
In this session, you will look at a few proofs and several applications of one of the most famous theorems in mathematics: the Pythagorean theorem. Proof is an essential part of mathematics, and what separates it from other sciences. Mathematicians start from assumptions and definitions, then follow logical steps to draw conclusions. If the assumptions are correct and the steps are indeed logical, then the result can be trusted and used to prove further results. When a result has been proved, it becomes a theorem.
For information on required and/or optional materials for this session, see Note 1.
Learning Objectives
In this session, you will learn how to do the following:
- Examine different formal proofs of the Pythagorean theorem
- Examine some applications of the Pythagorean theorem, such as finding missing lengths
- Learn how to derive and use the distance formula
Key Terms
Previously Introduced
Altitude: An altitude of a triangle is a line segment connecting a vertex to the line containing the opposite side and perpendicular to that side.
Perpendicular Bisector: The perpendicular bisector of a line segment is perpendicular to that segment and bisects it; that is, it goes through the midpoint of the segment, creating two equal segments.
Right Triangle: A right triangle is a triangle with one right (90°) angle.
Side-Angle-Side (SAS) Congruence: Side-angle-side (SAS) congruence states that if any two sides of a triangle are equal in length to two sides of another triangle and the angles bewteen each pair of sides have the same measure, then the two triangles are congruent; that is, they have exactly the same shape and size.
New in This Session
Converse: Converse means the “if” and “then” parts of a sentence are switched. For example, “If two numbers are both even, then their sum is even” is a true statement. The converse would be “If the sum of two numbers is even, then the numbers are even,” which is not a true statement.
Coordinates: Points are geometric objects that have only location. To describe their location, we use coordinates. We begin with a standard reference frame (typically the x- and y-axes). The coordinates of a point describe where it is located with respect to this reference frame. They are given in the form (x,y) where the x represents how far the point is from 0 along the x-axis, and the y represents how far it is from 0 along the y-axis. The form (x,y) is a standard convention that allows everyone to mean the same thing when they reference any point.
Hypotenuse: The hypotenuse in a right triangle is the side of the triangle that is opposite to the right angle.
Pythagorean Theorem: The Pythagorean theorem states that if you have a right triangle, then the square built on the hypotenuse is equal to the sum of the squares built on the other two sides. a2 + b2 = c2.
Theorem: A theorem in mathematics is a proven fact. A theorem about right triangles must be true for every right triangle; there can be no exceptions. Just showing that an idea works in several cases is not enough to make an idea into a theorem.
Notes
Note 1
Materials Needed:
- scissors
- graph paper (at least 10 pages)
