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.

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

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

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

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

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

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

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

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

Find missing values a and ba= 3 b=43 5 4 5

33

55

44

aabb

Find missing value for y

• 3 = y• y 4/3

4/34/333

yy