Download - Geometric mean
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Geometric mean
When an altitude is drawn from the vertex of a right triangle's 90 degree angle to its hypotenuse, it splits the
triangle into 2 right triangles that exhibit a special relationship.
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Theorem 50-1
• If the altitude is drawn to the hypotenuse of a right triangle, then the 2 triangles formed are similar to the original triangle
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CAUTION !!!
• Theorem 50-1 is only true if the altitude of the right triangle has an endpoint on the hypotenuse, not on the triangle's legs
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Identifying similar right triangles
• Find RS and RQ,- Tri PQR is similar to Tri PSQ is similar to Tri RSQ
PP
QQRR
SS
44
33
55
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Finding geometric mean
• Sometimes , the means of a proportion are equal to one another.
• This is a special kind of proportion that can be used to find the geometric mean of 2 numbers
• The geometric mean for positive numbers a and b, is the positive number x , such that
a
x
x
b
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Another way to state geometric mean
• The geometric mean of a and b is equal to the square root of the product of a and b, since
• ab = x2
a
x
x
b
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Find the geometric mean
• Find the geometric mean of 3 and 12
• Find the geometric mean of 4 and 16• Find the geometric mean of 2 and 9 in simplified
radical form.• Find the geometric mean of 5 and 11 to nearest
tenth.
3
12x
x
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Corollary 50-1-1
• If the altitude is drawn to the hypotenuse of a right triangle, then the length of the altitude is the geometric mean between the segments of the hypotenuse
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Corollary 50-1-2
• If the altitude is drawn to the hypotenuse of a right triangle, then the length of a leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is closer to that leg.
• AZ = YZ• YZ XZ
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Find missing values a and ba= 3 b=43 5 4 5
33
55
44
aabb
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Find missing value for y
• 3 = y• y 4/3
4/34/333
yy
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