Pythagorean Relationship Chapter 6 Pythagorean Relationship
6.1 Right Triangles The most commonly used triangle is the “right triangle” It is especially useful to use them in construction as they provide significant structural support. This is a right triangle with the terms we will use in this unit.
What is the hypotenuse? In a right triangle there is always a 90 degree angle. Opposite the right angle is always the longest side of the triangle. The longest side is always called the hypotenuse. Both of the shorter edges are called sides or “legs”
Examples of right triangles with lenghts
6.2 Pythagorean Relationship A relationship which allows us to find a missing side of a right angle triangle. When two sides are known the missing side can be found. This is very useful since it saves you having to measure the unknown side.
Solve for the hypotenuse in each right triangle
Example 2
Example 3
Practice Problems ● pp. 296-297: #1-7 If side lengths are given as fractions convert them to decimals to do the problem.
Finding the length of a leg when hypotenuse is given The Pythagorean theorem can be used to find any side of a triangle. You still need to know two sides of the triangle. This will let you find the third side. When finding a leg the equation used changes. b2 = c2 - a2
Finding the side length of a right triangle
Example 2
Some triangles don’t use “a” “b” “c”
Practice finding the leg of a right triangle • pp. 299-300: #1-6
Review of both Pythagorean triangles
Mixed practice finding hypotenuse and side lenghts • pp. 301-302: #1-9 (word problems) In this problem set be careful which side they are asking you to find. Finding Hypotenuse: a2 + b2 = c2 Finding a side length: b2 = c2 - a2
6.3 Using the Pythagorean Relationship It is very common to see triangles in modern day construction. They provide strong support to structures and are easy to perform calculations with. Square: Is a term used to refer to a corner that is exactly 90⁰. (Right angle) In this section the word “square” is not talking about the shape. It is referring to the fact that the corner or edge is exactly 90⁰. **This is usually desired in construction.**
Using Pythagorean theory to find if a triangle has a right angle in it. If all sides of a triangle are given you can use the formula a2 + b2 = c2 If the side lengths squared add up to the hypotenuse length then the triangle does contain a right angle. If the side lengths squared do not add up to the hypotenuse length then the triangle does not contain a right angle.
Checking a triangle for a right angle Use Pythagorean theory to determine if the following triangle contains a right angle.
Example 2 Does this triangle contain a right angle?
Example 3: Sometimes a conversion is needed. Suppose you are building a storage box. If the box square if it is built with the following measurements? (12 in = 1ft)
Assigned Practice P. 306-307 # 1-6 (in #3 we will use a2 + b2 = c2 There are a few questions where unit conversions will be needed. (1000m = 1km) (12 in = 1 ft)