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Area of a TriangleArea of a Triangle
A
B
12cm C
10cm
Example 1 : Find the area of the triangle ABC.
50o
(i) Draw in a line from B to AC
(ii) Calculate height BD
D
o BDSin50 =
10oBD = 10 Sin50 = 7.66
2
1
2
0.5 12 7.66 46
Area base height
cm
(iii) Area
7.66cm
Area of a TriangleArea of a Triangle
Q
P
20cm R
12cm
Example 2 : Find the area of the triangle PQR.
40o
(i) Draw in a line from P to QR
(ii) Calculate height PS
S
o PSSin40 =
10oPS = 12 Sin40 = 7.71
2
1
2
0.5 20 7.71 77.1
Area base height
cm
(iii) Area
7.71cm
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Know the formula for the Know the formula for the area of any triangle.area of any triangle.
1. To explain how to use the Area formula for ANY triangle.
Area of ANY TriangleArea of ANY Triangle
2.2. Use formula to find area of Use formula to find area of any triangle given two any triangle given two length and angle in length and angle in between.between.
General Formula forGeneral Formula forArea of ANY TriangleArea of ANY Triangle
Consider the triangle below:
Ao Bo
Co
ab
c
h
Area = ½ x base x height 1
2A c h
What does the sine of Ao equal
sin o hA
b
Change the subject to h. h = b
sinAoSubstitute into the area formula
1sin
2oA c b A
1sin
2oA bc A
Area of ANY TriangleArea of ANY Triangle
A
B
C
A
aB
b
Cc
The area of ANY triangle can be found by the following formula.
sin1
Area= ab C2
sin1
Area= ac B2
sin1
Area= bc A2
Another version
Another version
Key feature
To find the areayou need to knowing
2 sides and the angle in between (SAS)
Area of ANY TriangleArea of ANY Triangle
A
B
C
A
20cmB
25cm
Cc
Example : Find the area of the triangle.
sinC1
Area= ab2
The version we use is
30o
120 25 sin30
2oArea
210 25 0.5 125Area cm
Area of ANY TriangleArea of ANY Triangle
D
E
F
10cm
8cm
Example : Find the area of the triangle.
sin1
Area= df E2
The version we use is
60o
18 10 sin 60
2oArea
240 0.866 34.64Area cm
What Goes In The Box What Goes In The Box ??
Calculate the areas of the triangles below:
(1)
23o
15cm
12.6cm
(2)
71o
5.7m
6.2m
A =36.9cm2
A =16.7m2
Key feature
Remember (SAS)
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Know how to use the sine Know how to use the sine rule to solve REAL LIFE rule to solve REAL LIFE problems involving problems involving lengths.lengths.
1. To show how to use the sine rule to solve REAL LIFE problems involving finding the length of a side of a triangle .
Sine RuleSine Rule
C
B
A
Sine RuleSine Rule
a
b
c
The Sine Rule can be used with ANY triangle as long as we have been given enough information.
Works for any Triangle
a b c= =
SinA SinB SinC
Deriving the rule
B
C
A
b
c
a
Consider a general triangle ABC.
The Sine Rule
Draw CP perpendicular to BA
P
CPSinB CP aSinB
a
CP
also SinA CP bSinAb
aSinB bSinA
aSinBb
SinA
a bSinA SinB
This can be extended to
a b cSinA SinB SinC
or equivalentlySinA SinB SinCa b c
Calculating Sides Calculating Sides Using The Sine RuleUsing The Sine Rule
10m
34o
41o
a
Match up corresponding sides and angles:
sin 41oa
10
sin 34o
Rearrange and solve for a. 10sin 41
sin34
o
oa 10 0.656
11.740.559
a m
Example 1 : Find the length of a in this triangle.
A
B
C
sin sin sino
a b c
A B C
Calculating Sides Calculating Sides Using The Sine Using The Sine
RuleRule
10m133o
37o
d
sin133od
10
sin 37o
10sin133
sin 37
o
od
10 0.731
0.602d
=
12.14m
Match up corresponding sides and angles:
Rearrange and solve for d.
Example 2 : Find the length of d in this triangle.
C
D
E
sin sin sino
c d e
C D E
What goes in the Box What goes in the Box ??
Find the unknown side in each of the triangles below:
(1) 12cm
72o
32o
a
(2)
93o
b47o
16mm
a = 6.7cm b =
21.8mm
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Know how to use the sine Know how to use the sine rule to solve problems rule to solve problems involving angles.involving angles.
1. To show how to use the sine rule to solve problems involving finding an angle of a triangle .
Sine RuleSine Rule
Calculating Angles Calculating Angles
Using The Sine Using The Sine RuleRule
Example 1 :
Find the angle Ao
A
45m
23o
38m
Match up corresponding sides and angles:
45
sin oA 38
sin 23o
Rearrange and solve for sin Ao
45sin 23sin
38
ooA = 0.463 Use sin-1 0.463 to find Ao
1sin 0.463 27.6o oA
sin sin sin
a b c
A B C
B
C
Calculating Angles Calculating Angles
Using The Sine Using The Sine RuleRule
143o
75m
38m
X
38
sin oX
75
sin143o
38sin143sin
75
ooX = 0.305
1sin 0.305 17.8o oX
Example 2 :
Find the angle Xo
Match up corresponding sides and angles:
Rearrange and solve for sin Xo
Use sin-1 0.305 to find Xo
Y
Z
sin sin sin
x y z
X Y Z
What Goes In The Box What Goes In The Box ??
Calculate the unknown angle in the following:
(1)
14.5m
8.9m
Ao
100o (2)
14.7cm
Bo
14o
12.9cm
Ao = 37.2o
Bo = 16o