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Similar Triangles
Congruent Triangles
Similar Triangles
∆ KLM and ∆ TUV are similar
- corresponding angles are equal
- ratios of corresponding sides are equal
- ratio of the areas is equal to the ratio of the squares of corresponding sides
Example 1: ∆ BCD ~ ∆ EFG
Find the measures of c and e.
Example 2: Are the triangles similar?
Find x and y.
p. 322 #1ade, 2, 7, 9, 12
Primary Trigonometric Ratios
Sine, Cosine, Tangent
How do the buttons on your calculator work? What do they do?
Draw a right angled triangle that has a 30o angle. Measure all three sides.
1) Calculate
2) Calculate
3) Calculate
We have discovered three equations...
Example: Write the sine, cosine and tangent ratio for angle A. Then calculate angle A.
p. 330 #2,3,4,6
p. 338 #2 - 5
p. 344 #2 - 5
The Primary Trigonometric Ratios
Example 1: Use the trig ratios to find the required angle. Round to the nearest hundredth.
Example 2: Use the trig ratios to find the side length x. Round to the nearest hundredth.
Example 3: Solve the triangle.
Warm up
For safety, the angle a ladder makes with the ground should be between 60 o and 75o.
The base of an 8m ladder is placed 1.5m from the wall.
a) Is the ladder safe?
b) Determine the maximum and minimum distances the base of the ladder can be placed
from the wall safely.
Solve Problems Involving Right Triangles
Definitions:
Angle of Elevation
Angle of Depression
Example 1: Kim and Yuri live in apartment buildings that are 30 m apart. The angle of depression
from Kim’s balcony to where Yuri’s building meets the ground is 40o. The angle of elevation from
Kim’s balcony to Yuri’s balcony is 20o.
(a)
(b)
How high is Kim’s balcony above the ground, to the nearest metre?
How high is Yuri’s balcony above the ground, to the nearest metre?
Example 2: Using the diagram below, find the height of the cliff, to the nearest metre.
Entertainment:
p. 332 #11 - 17
p. 339 #10 - 13, 15
p. 344 #9 - 12
Review
1) The triangles below are similar.
ΔABC~ ΔDEF
Determine the missing side lengths.
2) Find the missing side length.
3) Find the missing angle.
p. 380 #1
p. 381 #4, 5, 7 - 10 (part a for all)
p. 384 #11ac, 12 -14
p. 392 #5
Warm up
Find the length of BD
Problems with Two Right Triangles
Example 1: Find the length of KL
Example 2: Cloud Height at Night
Aircraft from small airports can only fly if the cloud height is 300 m or higher. To determine the
cloud height at night, many small airports have a spotlight that shines on the clouds. The angle the
light beam makes with the ground is 70o. An observer, located on the ground 300m from the light,
measures the angle of elevation of this point is 60o, and the light and the observer are on opposite
sides of the point. Find the cloud height to the nearest metre.
p. 355 #1,2,3,6,7,8,11,14,20
The Sine Law
For right angled triangles we can use...
For non-right angled triangles...
1) Make a triangle (any size any shape)
2) Measure all the sides and all the angles
3) Now calculate...
Examples
1) Find angle R
Practice
p. 366 #1b-4b,6,7,9,10,12,13
2) Find a
Warm up
Determine the area of the triangle.
Proof of the Sine Law
Ambiguous Case of the Sine Law
Draw two different triangles, both with the following dimensions.
Then solve them both.
b = 7 cm, c = 10 cm, B = 30o
If angle A and side length b are both fixed in triangle ABC...
Create 1 triangle:
a=?
Create no triangles:
a=?
Create two triangles:
a=?
The Cosine Law
Example 1: Find side b
Example 2: Solve the triangle
Angle version of the cosine law
Practice: p. 373 # 1, 2, 3ab, 13
Warm up
Determine the area of this triangle.
Applications
In 700 BC, engineers on the Greek Island of Samos constructed a tunnel through Mount
Kastron to bring water from one side of the mountain to a city on the other side. The
digging teams started at opposite ends of the mountain and met in the centre.
A surveyor decides to find out the length of the tunnel. They stand far back from the
mountain and make the measurements shown below.
p. 374 #5 - 10, 15, 17