Honors Geometry Warm-up 1/30Ashwin is watching the Super Bowl on a wide screen TV with dimensions 32” by 18” while Emily is watching it on an old square TV with the same area. What are the dimensions of Emily’s TV?

Answer: 24” X 24”

Geometric Mean• Square root of the product ()• For positive numbers a and b, the

geometric mean is the positive number x where the proportion a:x = x:bor is true.

• Cross products: x2 = ab or .• > 2 numbers:

Example 1: Find the geometric mean between each pair of numbers.

a. 2 and 50

b. 25 and 7

Example 1C: True or False?The geometric mean of two distinct positive numbers can be greater than the average of

the two numbers.a, a + n (where n is the difference between the two numbers)

=

Altitude of a Triangle: Consider the right triangle ABC with altitude drawn from the right angle B to the hypotenuse .

Which triangles are similar to the big triangle ABC?

mBDA = 90.00

mABC = 90.00

D

A

B

C

∆ 𝐴𝐵𝐶 ∆ 𝐴𝐷𝐵𝑎𝑛𝑑∆ 𝐴𝐵𝐶 ∆𝐵𝐷𝐶

Theorem 7.1If the altitude is drawn from the vertex of the

right angle of a right triangle to its hypotenuse, then the two triangles formed are similar to the given triangle and to each other.

Example: mBDA = 90.00

mABC = 90.00

D

A

B

C

Proportional SidesLook at the smaller triangles:

We can say Since BD = DB,

Hence, the altitude is the geometric mean of the segments of the hypotenuse.

D

A

B

C

C

D

Theorem 7.2The measure of an altitude drawn from the

vertex of the right triangle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.

Example : BD is the geometric mean of AD and DC

DA

B

C

More Proportional Sides

DA

B

C

∆ 𝐴𝐵𝐶 ∆ 𝐴𝐷𝐵 ∆𝐵𝐷𝐶

𝐴𝐵𝐴𝐷

=𝐴𝐶𝐴𝐵

𝑜𝑟𝐵𝐶𝐷𝐶

=𝐴𝐶𝐵𝐶

Theorem 7.3 If the alt. is drawn from the vertex of the right

angle of a right triangle to its hyp., then the measure of a leg of the triangle is the geometric mean between the measures of the hyp. and the segment of the hyp. adjacent to that leg.

Example: AB is the geometric mean of AD and AC; BC is the geometric mean of DC and AC

DA

B

C

Example 2: In triangle ABC, BD=6 and AD=27. Find CD.

D

C

A

B

Example 3: Mrs. Clark is constructing a kite for her son. She has to arrange perpendicularly two support rods, the shorter of which is 27 inches

long. If she has to place the short rod 7.25 inches from one end of the long rod in order to form two

right triangles with the kite fabric, what is the length of the long rod?

Long rod = 32.39 in

Example 4: Find c and d in triangle JKL.

5

cd10

M

J

KL