March 8- Tuesday
Find the geometric mean of each pair of numbers:
1. ½ and 1 = 𝟏
2.
2 and 8 = 𝟐
3. 1 and 1000 = 𝟏𝟎 𝟏𝟎
4. a) Write a similarity statement relating the three triangles in the diagram.
PQR 
PSQ   QSR
b) If PS = 21 and SR = 4, find the lengths of 𝑃𝑄, 𝑄𝑆, and 𝑄𝑅.
𝑸𝑹 = 𝟏𝟎
𝑷𝑸 = 𝟓 𝟐𝟏
𝑸𝑺 = 𝟐 𝟐𝟏
Right Triangle Similarity Theorem
The altitude to the hypotenuse of a right  divides the  into
two s that are similar to the original  and to each other.
Corollary 1:
Geometric Mean (Altitude) Theorem
9-4 SIMILARITY IN
RIGHT TRIANGLES
The length of the altitude to the hypotenuse of a right  is the
geometric mean of the lengths of the segments of the hypotenuse.
Corollary 2:
Geometric Mean (Leg) Theorem
The altitude to the hypotenuse of a right  separates the hypotenuse so that the
length of each leg of the  is the geometric mean of the length of the hypotenuse
and the length of the segment of the hypotenuse adjacent to the leg.
9-4 Similarity in Right s
9-4 SIMILARITY
IN
RIGHT TRIANGLES
Real – World Problem Applications
PROBLEM #1 (Civil Engineering) Study the plan below.
A service station will be built on the highway, and a road will connect it with Cray.
a. How far from Blare should the service station be located so that the proposed
road will be perpendicular to the highway?
b. How long will the new road be?
Solution:
Let x = distance of the Service Station from Blare
𝒙 + 𝟑𝟐 30
𝒙 − 𝟏𝟖 𝒙 + 𝟓𝟎 = 𝟎
=
30
𝒙
𝒙 − 𝟏𝟖 = 𝟎 𝒙 + 𝟓𝟎 = 𝟎
𝒙 𝒙 + 𝟑𝟐 = 𝟑𝟎𝟐
𝒙 = 𝟏𝟖
𝒙 = −𝟓𝟎
𝒙𝟐 + 𝟑𝟐𝒙 = 𝟗𝟎𝟎
𝐚𝐧𝐬𝐰𝐞𝐫: 𝐚) 𝒙 = 𝟏𝟖 𝒎𝒊
𝟐
𝒙 + 𝟑𝟐𝒙 − 𝟗𝟎𝟎 = 𝟎
Let y = length of the new road
𝟑𝟐
𝒚
=
𝒚
𝟏𝟖
𝒚𝟐 = 𝟑𝟐 𝟏𝟖
𝒚=
𝟑𝟐 𝟏𝟖
𝒚 = 𝟐𝟒
𝐚𝐧𝐬𝐰𝐞𝐫: 𝐛) 𝒚 = 𝟐𝟒 𝒎𝒊
9-4 SIMILARITY
IN
RIGHT TRIANGLES
Real – World Problem Applications
PROBLEM #2 (The Totem Pole) To estimate the height of a totem pole, Jorge uses a small square of
plastic. He holds the square up to his eyes and walks backward from the pole. He stops when the bottom
of the pole lines up with the bottom edge of the square and the top of the pole lines up with the top edge
of the square. Jorge’s eye level is about 2 m from the ground. He is about 3 m from the pole. Which is the
best estimate for the height of the totem pole?
9-4 SIMILARITY
IN
RIGHT TRIANGLES
Real – World Problem Applications
PROBLEM #3 (STEM)
The architect’s side view drawing of a saltbox-style house shows a post that supports
the roof ridge. The support post is 10 ft tall. How far from the front of the house is the
support post positioned?
9-4 SIMILARITY
IN
RIGHT TRIANGLES
PROBLEM #4 (1F. Analyze Mathematical Relationships)
𝐶𝐷 is the altitude to the hypotenuse of right ABC. The coordinates of A, D, and B
are (4, 2), (4, 6), and (4, 15), respectively. Find all possible coordinates of point C.
9-4 SIMILARITY
IN
RIGHT TRIANGLES
Exit Ticket
The roof of a house forms a right angle, with each side of the roof measuring
28 ft in length. Find the width and height of the roof.