Name: _________________________________________________
Geometric Mean There are many ways to determine measures on a right triangle. Geometric mean is another method of doing this. It works with something called an altitude,β which is a segment connecting the right angle to the hypotenuse at another right angle. There are 2 formulas you will use to solve using the geometric mean.
To determine the legs (sides not the hypotenuse), use:
πππ! = (ππππ ππ π‘ ππππ‘ ππ π‘βπ βπ¦πππ‘πππ’π π)(βπ¦πππ‘πππ’π π)
To determine the altitude (segment going through) use:
πππ‘ππ‘π’ππ! = (πππ ππππ‘ ππ π‘βπ βπ¦πππ‘πππ’π π)(ππ‘βππ ππππ‘) or
part closest to x: 1 part closest to z: 2 whole hypotenuse: π + π = π
πππ ! = (ππππ ππ π‘)(π€βπππ) πππ ! = (ππππ ππ π‘)(π€βπππ) π₯! = 1 3 π§! = 2 3 π₯! = 3 π§! = 6 π₯! = 3 π§! = 6 π₯ = 3 π§ = 6
one part: 1 other part: 2 πππ‘ππ‘π’ππ ! = (πππ ππππ‘)(ππ‘βππ ππππ‘)
π¦! = 1 2 π¦! = 2 π¦! = 2 π¦ = 2
Determine the value of each variable. EXAMPLE
part closest to x = 4 part closest to z = 2
whole hypotenuse = 4 + 2 = 6 The legs: (πππ)! = (ππππ ππ π‘)(βπ¦πππ‘πππ’π π)
π₯! = 4 6 π₯! = 24 π₯! = 24
π₯ = 4 6 = 2 6
π§! = 2 6 π§! = 12 π§ = 12
π§ = 4 3 = 2 3
The altitude: (πππ‘ππ‘π’ππ)! = (πππ ππππ‘)(π‘βπ ππ‘βππ)
π¦! = 4 2 π¦! = 8 π¦! = 8
π¦ = 4 2 = 2 2
EXAMPLE
part closest to x = 6 part closest to y = 3
whole hypotenuse = 6 + 3 = 9 The legs: (πππ)! = (ππππ ππ π‘)(βπ¦πππ‘πππ’π π)
π₯! = 6 9 π₯! = 54 π₯! = 54
π₯ = 9 6 = 3 6
π¦! = 3 9 π¦! = 27 π¦! = 27
π¦ = 9 3 = 3 3
The altitude: (πππ‘ππ‘π’ππ)! = (πππ ππππ‘)(π‘βπ ππ‘βππ)
π§! = 3 6 π§! = 18 π§! = 18
π§ = 9 2 = 3 2
EXAMPLE
part closest to x = 5 part closest to z = 5
whole hypotenuse = 5 + 5 = 10 The legs: (πππ)! = (ππππ ππ π‘)(βπ¦πππ‘πππ’π π)
π₯! = 5 10 π₯! = 50 π₯! = 50
π₯ = 25 2 = 5 2
π§! = 5 10 π§! = 50 π§ = 50
π§ = 25 2 = 5 2 The altitude: (πππ‘ππ‘π’ππ)! = (πππ ππππ‘)(π‘βπ ππ‘βππ)
π¦! = 5 5 π¦! = 25 π¦! = 25 π¦ = 5
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