y 6 5 x 4 2 z y -...
Transcript of y 6 5 x 4 2 z y -...
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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,” 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:
𝑙𝑒𝑔! = (𝑐𝑙𝑜𝑠𝑒𝑠𝑡 𝑝𝑎𝑟𝑡 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)(ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)
To determine the altitude (segment going through) use:
𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒! = (𝑜𝑛𝑒 𝑝𝑎𝑟𝑡 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)(𝑜𝑡ℎ𝑒𝑟 𝑝𝑎𝑟𝑡) or
part closest to x: 1 part closest to z: 2 whole hypotenuse: 𝟏 + 𝟐 = 𝟑
𝑙𝑒𝑔 ! = (𝑐𝑙𝑜𝑠𝑒𝑠𝑡)(𝑤ℎ𝑜𝑙𝑒) 𝑙𝑒𝑔 ! = (𝑐𝑙𝑜𝑠𝑒𝑠𝑡)(𝑤ℎ𝑜𝑙𝑒) 𝑥! = 1 3 𝑧! = 2 3 𝑥! = 3 𝑧! = 6 𝑥! = 3 𝑧! = 6 𝑥 = 3 𝑧 = 6
one part: 1 other part: 2 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 ! = (𝑜𝑛𝑒 𝑝𝑎𝑟𝑡)(𝑜𝑡ℎ𝑒𝑟 𝑝𝑎𝑟𝑡)
𝑦! = 1 2 𝑦! = 2 𝑦! = 2 𝑦 = 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: (𝑙𝑒𝑔)! = (𝑐𝑙𝑜𝑠𝑒𝑠𝑡)(ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)
𝑥! = 4 6 𝑥! = 24 𝑥! = 24
𝑥 = 4 6 = 2 6
𝑧! = 2 6 𝑧! = 12 𝑧 = 12
𝑧 = 4 3 = 2 3
The altitude: (𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒)! = (𝑜𝑛𝑒 𝑝𝑎𝑟𝑡)(𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟)
𝑦! = 4 2 𝑦! = 8 𝑦! = 8
𝑦 = 4 2 = 2 2
EXAMPLE
part closest to x = 6 part closest to y = 3
whole hypotenuse = 6 + 3 = 9 The legs: (𝑙𝑒𝑔)! = (𝑐𝑙𝑜𝑠𝑒𝑠𝑡)(ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)
𝑥! = 6 9 𝑥! = 54 𝑥! = 54
𝑥 = 9 6 = 3 6
𝑦! = 3 9 𝑦! = 27 𝑦! = 27
𝑦 = 9 3 = 3 3
The altitude: (𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒)! = (𝑜𝑛𝑒 𝑝𝑎𝑟𝑡)(𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟)
𝑧! = 3 6 𝑧! = 18 𝑧! = 18
𝑧 = 9 2 = 3 2
EXAMPLE
part closest to x = 5 part closest to z = 5
whole hypotenuse = 5 + 5 = 10 The legs: (𝑙𝑒𝑔)! = (𝑐𝑙𝑜𝑠𝑒𝑠𝑡)(ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)
𝑥! = 5 10 𝑥! = 50 𝑥! = 50
𝑥 = 25 2 = 5 2
𝑧! = 5 10 𝑧! = 50 𝑧 = 50
𝑧 = 25 2 = 5 2 The altitude: (𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒)! = (𝑜𝑛𝑒 𝑝𝑎𝑟𝑡)(𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟)
𝑦! = 5 5 𝑦! = 25 𝑦! = 25 𝑦 = 5
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