Transcript 8.1 notes for absent students

When the means of a proportion are the same number, that number is
called the geometric mean of the extremes. The geometric mean
between two numbers is the positive square root of their product.
𝑥 2 = ab
𝑥= 𝑎𝑏
Geometric Mean
) Find the geometric mean between 2 and 50.
The means of the proportion are missing. Start filling in:
𝑥
=
𝑥
The other two given numbers are the missing parts of the proportion:
2
𝑥
Cross multiply and solve: 𝑥 2 = 100
𝑥= 10
B) Find the geometric mean between 3 and 12.
TRY SOLVING ON YOUR OWN FIRST:
𝑥
3
𝑥
=
𝑥
𝑥
= 12
𝑥 2 = 36
𝑥= 6
𝑥
= 50
Altitude of a triangle: The altitude of a triangle is a segment drawn from a
vertex to the line containing the opposite side and perpendicular to the line
containing that side.
Geometric Means in Right Triangles: In a right triangle, the altitude drawn
from the vertex of the right angle to the hypotenuse forms two additional right
triangles. These three right triangles share a special relationship.
This is a right
Δ with an
altitude drawn
to the hypotenuse
This is the figure
we will see in the
rest of this
section.
Identify Similar Right Triangles
Write a similarity statement identifying the three
similar triangles in the figure.
• By definition of similar polygons, you can write
proportions comparing the side lengths of these
triangles.
Method 1: use similar Δs
from ch 7 to compare
corresponding sides.
Small Δ: medium Δ
Large Δ : small Δ
Large Δ : medium Δ
• Notice that the circled relationships involve geometric
means. This leads to the next theorem.
WE DO NOT USE THIS METHOD, but still use the geometric mean pattern!!
Method 2: Set up one of THREE possible proportions
1) The altitude is the geometric mean,
Or 3) The long leg is the
geometric mean
2) The short leg is the geometric mean,
When the short or long leg is used for the geometric mean,
make sure you use the entire hypotenuse length in one of
the two remaining spots of the proportion.
Geometric Mean
To help remember the proportion set up, we use this story.
Meet Parachute Pete.
Pete can parachute down 1 of 3 possible paths.
Use worksheet
“Parachute Pete
Story” with the
next 3 slides
He can parachute
the short legHe can parachute
the altitutdeHe can parachute
the long leg-
The paths that Pete follows to the ground are the Geometric
Mean. These distances are placed diagonally in the
proportion!
Therefore you can write 3
proportions to find missing
side lengths!
If he “parachutes” down a
leg, fill in the geometric
mean. Then, once he
“lands” he can visit the
close city or the far city.
(WATCH COLOR CODED
ANIMATION)
If he “parachutes” down the
altitude, fill in the geometric
mean. Then, once he
“lands” he can visit the left
city or the right city.
(WATCH COLOR CODED
ANTIMAION)
BC
B
If he “parachutes” down a
leg, then once he “lands”
he can visit the close city
or the far city. (WATCH
COLOR CODED
ANIMATION)
A
CD
So the path he parachutes
is the geometric mean.
The story continues with
him landing on the ground
and having one plane ticket
to visit another city.
“Parachute”, then fill in the
geometric mean of the
proportion.
C
D
=
BC
AD
CA
BD
=
BD
DC
Again, notice how the “far city” uses the entire
hypotenuse length when using a leg as the
geometric mean.
AD
BA
=
BA
AC
Parachute Pete, Pg. 36 in WB
*Pete always parachutes from theB RIGHT ANGLE of the LARGE
C
triangle.
C
Pete will start at C.
A
D
B
Pete will start at A.
A
D
*The path he travels is the GEOMETRIC MEAN (so it is used TWICE
in the proportion).
*Then he visits TWO cities: (these are the other two blanks in the
proportion).
If the path he traveled was the ALTITUDE (middle) path, then he visits
the LEFT city or the RIGHT city.
If the path he traveled was an OUTSIDE path, then he visits the
CLOSE city or the FAR city.
Parachute Pete, Pg. 37 in WB
For these problems,
use the highlighted segment
as Pete’s path to parachute.
A
C
B
BA
PB
PB
=
BC
A
C
B
AB
PA
=
PA
AC
A
C
B
CB
=
PC
PC
CA
Back to the “Parachute Pete Story” worksheet
TRY THESE ON YOUR OWN, THEN USE ANIMATION
TO GET ANSWERS TO APPEAR
Set up the proportion for each picture. Use the bold segment as Pete’s Path.
W
1.
𝑂𝐿
𝑊𝐿
=
𝑊𝐿
𝐿𝐹
L
O
2.
P
A
F
T
R
3.
𝑃𝐴 𝑅𝑃
=
𝑅𝑃 𝑃𝑇
𝐴𝑌 𝐷𝐴
=
𝐷𝐴 𝐴𝑆
D
4.
C
𝐿𝐴 𝐼𝐿
=
𝐼𝐿 𝐿𝐶
A
A
Y
S
I
L
“Parachute Pete Story” worksheet
TRY THESE ON YOUR OWN, THEN USE
ANIMATION
TO GET ANSWERS TO APPEAR
Set up two proportion for each picture. Use SEGMENT NAMES in the first;
Numbers and variables in the second. Solve each proportion for x.
N
K
5.
x
T
9
I
E
4
𝑂𝐿
𝑊𝐿
=
𝑊𝐿
𝐿𝐹
9
𝑥
=
𝑥
4
x
6.
𝑁𝑅 𝑇𝑁
=
𝑇𝑁 𝑁𝑈
4
T
R
4
𝑥
=
𝑥
9
5
U
x =6
7.
8.
C
9
E
7
H
x =6
x
S
𝐸𝐶 𝐻𝐸
=
𝐻𝐸 𝐸𝑆
9
7
=
7
𝑥
x = 5.4
C
3
S
8
E
𝑆𝐸 𝐻𝑆
=
𝐻𝑆 𝑆𝐶
x
H
8
𝑥
=
𝑥
11
x = 2 22
Remember simplifiying radicals? See next slide
for review
Simplifying Radicals (Square Roots)
- No Decimal Answers when you solve by
square rooting (unless there is a decimal
under the square root).
- Decimals are OK when you solve using
division!
x = 88
8
11
2
4
2
2 2 ∙ 11 = 2 22
Pick any 2 numbers that multiply to get 88.
Look for pairs of numbers at the bottom
2 of each branch.
One number from each pair goes on
the outside of the radical.
2
Any numbers without a pair get
multiplied on the inside of the radical.
Use Geometric Mean with Right Triangles
Back to WB Pg. 37!!!
Find c, d, and e.
PETE
When solving, sometimes it will matter
what variable you need to find first.
If you set up a proportion and more than
one variable is in that proportion, try another
path.
To solve for c: 24
𝑐
=
𝑐
30
c = 720 = 12 5
To solve for d:
𝑒
To solve for e: 6
=
𝑒 24
e = 144 = 12
6
𝑑
=
𝑑 30
d = 180 = 6 5
Find e to the nearest tenth.
𝑒
To solve for e: 16
=
𝑒
20
e = 320 = 𝟏𝟕. 𝟗
(e = 8 5)
PETE
Indirect Measurement
KITES Ms. Alspach is constructing a kite for her
son. She has to arrange two support rods so that
they are perpendicular. The shorter rod 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?
The long rod will be 7.25 + x, so
Parachute to solve for x.
13.5
7.25
x
7.25 13.5
=
13.5
𝑥
x = 25.14
25.14 + 7.25 = 32.39 in
AIRPLANES A jetliner has a wingspan, BD, of
211 feet. The segment drawn from the front of the
plane to the tail,
at point E.
If AE is 163 feet, what is the length of the aircraft
to the nearest tenth of a foot?
The length of the plane will be 163 + x, so
Parachute to solve for x.
105.5
163
163
105.5
=
105.5
𝑥
x = 68.28
68.28 + 163 = 231.3 ft
x