8-1 Geometric Meanp. 537
You used proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles.
β’ Find the geometric mean between two numbers.
β’ Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse.
Geometric mean β when the means of a proportion are the same number.
ππ
=ππ
extremeextrememeanmean
The geometric mean between two numbers is the positive square room of their product.
Find the geometric mean between 2 and 50.
Answer: The geometric mean is 10.
Definition of geometric mean
Let x represent the geometric mean.
Cross products
Take the positive square root of each side.
Simplify.
Geometric Means in Right Triangles
In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two additional right triangles.The two triangles formed are similar to the original triangle and to each other.
Write a similarity statement identifying the three similar triangles in the figure.
Separate the triangles into two triangles along the altitude.Then sketch the three triangles, reorienting the smaller ones so that their corresponding angles and sides are in the same position as the original triangle.
Answer: So, by Theorem 8.1, ΞEGF ~ ΞFGH ~ ΞEFH.
A. ΞLNM ~ ΞMLO ~ ΞNMO
B. ΞNML ~ ΞLOM ~ ΞMNO
C. ΞLMN ~ ΞLOM ~ ΞMON
D. ΞLMN ~ ΞLMO ~ ΞMNO
Write a similarity statement identifying the three similar triangles in the figure.
A B
C
D
ab
x y
h
c
A B
CA
B
C
D
C
Da
a
bb
x
c
h
h y
hπ πππ‘ππ πππππππππ πππ
=ππ
=π₯h=hπ¦
hπ¦πππ‘πππ’π πhπ πππ‘ππ πππ
=ππ
=ππ₯
=πh
p. 538
Find c, d, and e.Since e is the measure of the altitude drawn to the hypotenuse of right ΞJKL, e is the geometric mean of the lengths of the two segments that make up the hypotenuse, JM and ML.
Geometric Mean(Altitude) TheoremSubstitutionSimplify.
Geometric Mean (Leg) TheoremSubstitution
Use a calculator to simplify.
Since d is the measure of leg JK, d is the geometric mean of JM, the measure of the segment adjacent to this leg, and the measure of the hypotenuse JL.
Geometric Mean(Leg) Theorem
Substitution
Use a calculator tosimplify.
Answer: e = 12, d β 13.4, c β 26.8
Since c is the measure of leg KL, c is the geometric mean of ML, the measure of the segment adjacent to KL, and the measure of the hypotenuse JL.
Top Related