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MHF4U7
Unit 10 – Geometry of Triangles and Circles (Ch.10)
Mr. Kwok
10A Right-Angled Triangles
Primary Trigonometric Ratios:
sin 𝑥 =
𝑜𝑝𝑝
ℎ𝑦𝑝
cos 𝑥 =
𝑎𝑑𝑗
ℎ𝑦𝑝
tan 𝑥 =
𝑜𝑝𝑝
𝑎𝑑𝑗
Two people are trying to measure the width of a river. There is no bridge across the river, but they have instruments for
measuring lengths and angles. When they stand 50 m apart on the same side of the river, at points A and B, the person at A
measures that angle between line (AB) and the line from A to the tower on the other side of the river is 25.6°. The person
at B finds the corresponding angle to be 28.3°, as shown in the diagram. Calculate the width of the river.
Page 1
MHF4U7
Unit 10 – Geometry of Triangles and Circles (Ch.10)
Mr. Kwok
10B Sine Law
Sine Law:
𝑠𝑖𝑛 𝐴
𝑎
=
𝑠𝑖𝑛 𝐵
𝑏
=
𝑠𝑖𝑛 𝐶
𝑐
The Sine Rule is used to solve problems involving triangles given:
 Two angles and one side
 Two sides and a non-included angle
1.
Find x.
Ambiguous case of Sine Law occurs when an angle and 2 consecutive sides are given.
Since there is no angle given between the 2 consecutive sides, nothing prevents side
A from swinging around.
B
As a result, it’s possible to have 2 triangles from a single problem:
B
B
A
C
A
Triangle 1
To find ∠C of Triangle 2:
C
A
Triangle 2
Step
1.
2.
Find ∠C of Triangle 1 using Sine Law
∠C of Triangle 2 = 180° − ∠𝐶 𝑜𝑓 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 1
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MHF4U7
Unit 10 – Geometry of Triangles and Circles (Ch.10)
Mr. Kwok
2. Solve ∆𝐴𝐵𝐶. Find all possible solutions.
a)
𝐴 = 15°, 𝑎 = 10.2, 𝑏 = 17
b) 𝐴 = 29°, 𝑎 = 3, 𝑏 = 7.5
3. Find the measure of angle 𝐿 in triangle 𝐾𝐿𝑀 given that angle 𝐾 measures 56°, 𝐿𝑀 = 16.8𝑚, and 𝐾𝑀 = 13.5𝑚
Page 3
MHF4U7
Unit 10 – Geometry of Triangles and Circles (Ch.10)
Mr. Kwok
10C Cosine Law
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
B
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
c
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
A
a
b
C
The Cosine rule can be used to solve problems involving triangles given:
5 cm
17.2 m
6.4 m
48⁰
6 cm
21.3 m
1.
Find the length of 𝐵𝐶
2.
In triangle 𝐴𝐵𝐶, 𝐴𝐵 = 7 𝑐𝑚, 𝐵𝐶 = 5 𝑐𝑚, and 𝐶𝐴 = 8 𝑐𝑚. Find the measure of 𝐵𝐶𝐴
Find the angle B.
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MHF4U7
Unit 10 – Geometry of Triangles and Circles (Ch.10)
Mr. Kwok
10D Area of a Triangle
After learning how to calculate side lengths and angles of a triangle, we are also interested in finding the area of the
triangle.
1
𝐴𝑟𝑒𝑎 = (𝑏𝑎𝑠𝑒)(ℎ𝑒𝑖𝑔ℎ𝑡)
2
𝑎
𝜃
𝑏
What is height?
1.
𝐶
Find the area of the triangle ABC.
28°
11
15
𝐵
𝐴
Page 5
MHF4U7
Unit 10 – Geometry of Triangles and Circles (Ch.10)
Mr. Kwok
10E Trigonometry in 3D
A cuboid has sides of length 8 cm, 12 cm, and 15 cm.
The diagram shows diagonals of three of the faces.
a)
Find the lengths AB, BC, and CA
b) Find the size of the angle ACB
c)
Calculate the area of the triangle ABC
d) Find the length AD
Page 6
MHF4U7
Unit 10 – Geometry of Triangles and Circles (Ch.10)
10FG
Mr. Kwok
Length of an Arc & Area of a Sector
Identify the Minor Arc? Major Arc?
Arc Length
Ratio
Length : Angle
1.
Arc AB of a circle with radius 5cm subtends an angle of 0.6 at the centre.
a) Find the length of the minor arc AB.
b) Find the length of the major arc AB.
A sector is a part of a circle bounded by two radii and an arc.
Area of Sector
Ratio
Angle : Area
2.
A sector has radius 12 cm and angle 3 radians. Use radians to find its:
a) Arc length
b) Area
Page 7
MHF4U7
Unit 10 – Geometry of Triangles and Circles (Ch.10)
3.
A sector of a circle has perimeter 𝑝 = 12𝑐𝑚 and angle 𝜃 = 50° at the center. Find the area of the sector.
4.
Find the arc length and area of a sector of radius 5 cm and angle 2 radians.
5.
If a sector has radius 2𝑥 cm and arc length 𝑥 cm, show that its area is 𝑥 2 𝑐𝑚2 .
Mr. Kwok
Page 8
MHF4U7
Unit 10 – Geometry of Triangles and Circles (Ch.10)
Mr. Kwok
10H Triangles and Circles
Segment
A region of the circle that is cut off from the rest of the circle by a secant or a cord.
Formula:
Chord
A line segment joining two points on any curve.
Formula:
1.
Given the shaded area.
a) Find the perimeter.
b) Find the area.
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MHF4U7
2.
Unit 10 – Geometry of Triangles and Circles (Ch.10)
Mr. Kwok
The diagram shows two identical circle of radius 12 such that the center of one circle is on the circumference of the
other.
a)
Find the exact size of angle PC1 Q in radians.
b) Calculate the exact area of the shaded region.
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