what we do in life echoes in eternity.. 7.4 similarity in right triangles lt: to find and use...
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“What we do in life echoes in eternity.”
7.4 Similarity in Right Triangles
LT: To find and use relationships of similar right triangles.
Similarity in Right Triangles
Theorem 7-3: The altitude to the hypotenuse of a right triangle divides the triangles into two triangles that are similar to the original triangle and to each other.
Geometric Mean
Geometric Mean: The number x such that , where a, b, and x are positive numbers
a
xx
b
Find the geometric mean of 3 and 27.
Review: How do we find the arithmetic mean of 3 and 27?
Note:
x ab
Find the geometric mean of 4 and 18.
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers.
1. The geometric mean can give a meaningful "average" to compare two companies.
2. The use of a geometric mean "normalizes" the ranges being averaged, so that no range dominates the weighting.
3. The geometric mean applies only to positive numbers.[2]
4. It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment.
Purpose of the Geometric Mean
Geometric Mean
5.2 in 8.75in
6.75in
Corollary to Theorem 7-3: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse
Similarity in Right TrianglesFind the values of x and y in the following right triangle.
4 5
X Y Y
X
4 + 5
You Try One!!!Find the values of x and y in the following right triangle.
“You wasted $150,000 on an education you coulda got for $1.50 in late fees at the public library.”
7.4 Similarity in Right Triangles
HW (7.4) Pgs. 394-396: #1-21, 34, 35, 50, 51
Proof of Corollary to Theorem 7-3
A
C
D B
Statements Reasons
1.
2.
3.
1.
2.
3.
Right triangle, ABC, with
CD the altitude to the hypotenuse
AD
CDCD
DB
Given : Right triangle, ABC, with
CD the altitude to the hypotenuse
Prove : AD
CDCDDB
Altitude of rt. Δ to hypotenuse divides into 2 ~ Δs
Real World ConnectionAs Marla arrives at the lake from the parking lot, she reads a sign that says she is 320m from the dock. How far is Marla from the information center?
Kick it up a notch!Find the value of x in the following right triangle.
x1
2x - 1
Similarity in Right Triangles
m1m4 m7
m2 m6 m8
m3 m5 m9