Section 8.1 – 8.2 Geometric Mean Pythagorean Theorem Geometry Section 8.1 – 8.2 Geometric Mean Pythagorean Theorem

Today’s Agenda Geometric Mean in Right Triangles Pythagorean Theorem in Right Triangles

Geometric Mean A Geometric Mean is a kind of average. To find the Geometric Mean between two numbers, multiply them together and take the square root. Example: Find the Geometric Mean of 5 and 20.

Geometric Mean as a Proportion In a proportion if the means are equal, then that value is the geometric mean of the extremes: x represents the Geometric Mean between a and b. Example:

Geometric Mean This concept can used in Geometry. One particular use is when dealing with right triangles.

Geometric Mean We start with a Right Triangle

Geometric Mean Let’s draw its altitude.

Geometric Mean We’ve now formed 2 more triangles – 3 in all! What do these 3 triangles have in common?

Geometric Mean Let’s consider the original diagram C A B D

Geometric Mean We’ll put the others up for reference A C A A B D C D C

Geometric Mean Let’s label the sides A C A a b h d e A B a c D c a d C

Geometric Mean We can use similarity properties to set up proportions: b h d e A B a c D c a d C b h D C C B D B h b e

Geometric Mean To conclude, a, b, and h, can all be written as the Geometric Mean of two segments. a b h d e c

Geometric Mean Putting it in words: The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Example 5 20 x y z

Pythagorean Theorem: SECTION 8.2  

Example 1: If a = 9, b = 12, solve for c. YOU TRY IT!

Example #2: If a = 12, c = 20, solve for b. YOU TRY IT!

Example 3: If a = 6 and b = 15, solve for c Example 3: If a = 6 and b = 15, solve for c. (Answer in simplest radical form) YOU TRY IT:

Determine if a set of numbers can be the measures of sides of a triangle: USING the TRIANGLE INEQUALITY THEOREM: Example: 7, 15, 21 You try it!

Classify a triangle as Acute, Obtuse, or Right: