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TRIGONOMETRY
Math 10
Ms. Albarico
A. Modeling Situations Involving Right Triangles
B. Congruence and Similarity
5.1 Ratios Based on Right Triangles
1) Apply the properties of similar triangles.
2) Solve problems involving similar triangles and right triangles.
3) Determine the accuracy and precision of a measurement.
4) Solve problems involving measurement using bearings vectors.
Students are expected to:
Vocabulary
perpendicularparallelsidesangletrianglecongruentsimilardilatesailnavigateapproach
Introduction
Trigonometry is a branch of mathematics that uses triangles to
help you solve problems.
Trigonometry is useful to surveyors, engineers, navigators, and machinists
(and others too.)
Triangles Around Us
Review
• Types of Triangles (Sides)
• a) Scalene• b) Isosceles• c) Equilateral
Review
• Types of Triangles (Angles)
• a) Acute• b) Right• c) Obtuse
LABELING RIGHT TRIANGLES
• The most important skill you need right now is the ability to correctly label the sides of a right triangle.
• The names of the sides are:– the hypotenuse– the opposite side– the adjacent side
Labeling Right Triangles
• The hypotenuse is easy to locate because it is always found across from the right angle.
Here is the right angle...
Since this side is across from the right angle, this must be
the hypotenuse.
Labeling Right Triangles
• Before you label the other two sides you must have a reference angle selected.
• It can be either of the two acute angles.• In the triangle below, let’s pick angle B
as the reference angle.
A
B
C
This will be our reference angle...
Labeling Right Triangles
• Remember, angle B is our reference angle.
• The hypotenuse is side BC because it is across from the right angle.
A
B (ref. angle)
C
hypotenuse
Labeling Right Triangles
Side AC is across from our reference angle B. So it is labeled: opposite.
A
B (ref. angle)
Copposite
hypotenuse
Labeling Right Triangles
The only side unnamed is side AB. This must be the adjacent side.
A
B (ref. angle)
C
adjacenthypotenuse
opposite
Adjacent means beside or next to
Labeling Right Triangles
• Let’s put it all together.• Given that angle B is the reference
angle, here is how you must label the triangle:
A
B (ref. angle)
C
hypotenuse
opposite
adjacent
Labeling Right Triangles
• Given the same triangle, how would the sides be labeled if angle C were the reference angle?
• Will there be any difference?
Labeling Right Triangles
• Angle C is now the reference angle.• Side BC is still the hypotenuse since it
is across from the right angle.
A
B
C (ref. angle)
hypotenuse
Labeling Right Triangles
However, side AB is now the side opposite since it is across from angle C.
A
B
C (ref. angle)
oppositehypotenuse
Labeling Right Triangles
That leaves side AC to be labeled as the adjacent side.
A
B
C (ref. angle)adjacent
hypotenuseopposite
Labeling Right Triangles
Let’s put it all together. Given that angle C is the reference angle,
here is how you must label the triangle:
A
B
C (ref. angle)
hypotenuseopposite
adjacent
Labeling Practice
• Given that angle X is the reference angle, label all three sides of triangle WXY.
• Do this on your own. Click to see the answers when you are ready.
W X
Y
Labeling Practice
• How did you do?• Click to try another one...
W X
Y
hypotenuse
opposite
adjacent
Labeling Practice
• Given that angle R is the reference angle, label the triangle’s sides.
• Click to see the correct answers.R
ST
Labeling Practice
• The answers are shown below:
R
ST
hypotenuse
opposite
adjacent
Which side will never be the reference angle?
What are the labels?
Hypotenuse, opposite, and adjacent
The right angle
The Meaning of Congruence
A Congruent Figures
B Transformation and Congruence
C Congruent Triangles
Conditions for Triangles to be Congruent
Three Sides EqualA
Two Sides and Their Included Angle Equal
B
Two Angles and One Side EqualC
Two Right-angled Triangles with Equal Hypotenuses and Another Pair of Equal Sides
D
The Meaning of Similarity
A Similar Figures
B Similar Triangles
A Three Angles Equal
B Three Sides Proportional
Conditions for Triangles to be Similar
C Two Sides Proportional and their Included Angle Equal
Congruent Figures
1. Two figures having the same shape and the same size
are called congruent figures.
E.g. The figures X and Y as shown are congruent.
The Meaning of Congruence
Example
A)
2. If two figures are congruent, then they will fit exactly
on each other.
X Y
The figure on the right shows a symmetric
figure with l being the axis of symmetry.
Find out if there are any congruent figures.
The Meaning of Congruence
Therefore, there are two congruent figures.
The line l divides the figure into 2 congruent figures,
i.e. and are congruent figures.
Find out by inspection the congruent figures among the following.
The Meaning of Congruence
A B C D
E F G H
B, D ; C, F
Transformation and Congruence
‧ When a figure is translated, rotated or reflected, the
image produced is congruent to the original figure.
When a figure is enlarged or reduced, the image
produced will NOT be congruent to the original one.
The Meaning of Congruence
Example
B)
Note:
A DILATATION is a transformation which enlarges or reduces a shape but does not change its proportions.
SIMILARITY is the result of dilatation
" " means "is congruent to "≅
"~" is "similar to".
(a)(i) ____________
(ii) ____________
In each of the following pairs of figures, the red one is obtained by
transforming the blue one about the fixed point x. Determine
Index
The Meaning of Congruence
(i) which type of transformation (translation, rotation, reflection,
enlargement, reduction) it is,
(ii) whether the two figures are congruent or not.
Reflection
Yes
(b)
(i) ____________
(ii) ____________
(c)
(i) ____________
(ii) ____________
Index
The Meaning of Congruence
Translation
Yes
Enlargement
No
Back to Question
The Meaning of Congruence
Rotation
Yes
Reduction
No
Back to Question
(d)
(i) ____________
(ii) ____________
(e)
(i) ____________
(ii) ____________
Congruent Triangles
‧ When two triangles are congruent, all their corresponding
sides and corresponding angles are equal.
The Meaning of Congruence
Example
C)
E.g. In the figure, if △ABC △XYZ, C
A B
Z
X Y
then
∠A = ∠X,
∠B = ∠Y,
∠C = ∠Z,
AB = XY,
BC = YZ,
CA = ZX.
and
Name a pair of congruent triangles in the figure.
From the figure, we see that △ABC △RQP.
Trigonometry
Given that △ABC △XYZ in the figure, find the unknowns p,
q and r.
For two congruent triangles, their corresponding sides and angles are equal.
∴
∴ p = 6 cm , q = 5 cm , r = 50°
Trigonometry
Write down the congruent triangles in each of the following.
(a)
B
A
C
Y
X
Z
(a) △ABC △XYZ
(b) P
QR
S
T
U
(b) △PQR △STU
Trigonometry
Find the unknowns (denoted by small letters) in each of the
following.
Trigonometry
(a) x = 14 ,
(a) △ABC △XYZ
1314
15A
B
Cz
x
X
YZ
z = 13
(b) △MNP △IJK
M
35°98°
47°
P
N
I
j
i
K
J
(b) j = 35° , i = 47°
Three Sides Equal
Conditions for Triangles to be Congruent
Example
A)
‧ If AB = XY, BC = YZ and CA = ZX,
then △ABC △XYZ.
【 Reference: SSS 】
A
B C
X
Y Z
Determine which pair(s) of triangles in the following are congruent.
Conditions for Triangles to be Congruent
(I) (II) (III) (IV)
In the figure, because of SSS,
(I) and (IV) are a pair of congruent triangles;
(II) and (III) are another pair of congruent triangles.
Each of the following pairs of triangles are congruent. Which
of them are congruent because of SSS?
Conditions for Triangles to be Congruent
18
10
18
10A B
3
7
537
5
B
Two Sides and Their Included Angle Equal
Conditions for Triangles to be Congruent
Example
B)
‧ If AB = XY, ∠B = ∠Y and BC = YZ,
then △ABC △XYZ.
【 Reference: SAS 】
A
B C
X
Y Z
Determine which pair(s) of triangles in the following are congruent.
Conditions for Triangles to be Congruent
In the figure, because of SAS,
(I) and (III) are a pair of congruent triangles;
(II) and (IV) are another pair of congruent triangles.
(I) (II) (III) (IV)
In each of the following figures, equal sides and equal
angles are indicated with the same markings. Write down a
pair of congruent triangles, and give reasons.
Conditions for Triangles to be Congruent
(a) (b)
(a) △ABC △CDA (SSS)
(b) △ACB △ECD (SAS)
Two Angles and One Side Equal
Conditions for Triangles to be Congruent
Example
C)
1. If ∠A = ∠X, AB = XY and ∠B = ∠Y,
then △ABC △XYZ.
【 Reference: ASA 】
C
A B
Z
X Y
Two Angles and One Side Equal
Conditions for Triangles to be Congruent
Example
C)
2. If ∠A = ∠X, ∠B = ∠Y and BC = YZ,
then △ABC △XYZ.
【 Reference: AAS 】
C
A B
Z
X Y
Determine which pair(s) of triangles in the following are congruent.
Conditions for Triangles to be Congruent
In the figure, because of ASA,
(I) and (IV) are a pair of congruent triangles;
(II) and (III) are another pair of congruent triangles.
(I) (II) (III) (IV)
In the figure, equal angles are indicated
with the same markings. Write down a
pair of congruent triangles, and give
reasons.
Conditions for Triangles to be Congruent
△ABD △ACD (ASA)
Fulfill Exercise Objective
Identify congruent triangles from given diagram and given reasons.
Determine which pair(s) of triangles in the following are congruent.
1B_Ch11(55)Conditions for Triangles to be Congruent
In the figure, because of AAS,
(I) and (II) are a pair of congruent triangles;
(III) and (IV) are another pair of congruent
triangles.
(I) (II) (III) (IV)
In the figure, equal angles are indicated with the same
markings. Write down a pair of congruent triangles, and give
reasons.
1B_Ch11(56)Conditions for Triangles to be Congruent
A
B C
D
△ABD △CBD (AAS)
Two Right-angled Triangles with Equal Hypotenuses
and Another Pair of Equal Sides
Conditions for Triangles to be Congruent 1B_Ch11(57)
Example
D)
‧ If ∠C = ∠Z = 90°, AB = XY and BC = YZ,
then △ABC △XYZ.
【 Reference: RHS 】
A
B C
X
Y Z
Determine which of the following pair(s) of triangles are congruent.
Conditions for Triangles to be Congruent
In the figure, because of RHS,
(I) and (III) are a pair of congruent triangles;
(II) and (IV) are another pair of congruent triangles.
(I) (II) (III) (IV)
In the figure, ∠DAB and ∠BCD are
both right angles and AD = BC.
Judge whether △ABD and △CDB
are congruent, and give reasons.
Conditions for Triangles to be Congruent
Yes, △ABD △CDB (RHS)
Similar Figures
The Meaning of Similarity
Example
A)
1. Two figures having the same
shape are called similar figures.
The figures A and B as shown is
an example of similar figures.
2. Two congruent figures must be also similar figures.
3. When a figure is enlarged or reduced, the new figure is
similar to the original one.
Find out all the figures similar to figure A by inspection.
D, E
The Meaning of Similarity
A B C D E
Similar Triangles
Example
B)
1. If two triangles are similar, then
i. their corresponding angles are equal;
ii. their corresponding sides are
proportional.
The Meaning of Similarity
2. In the figure, if △ABC ~ △XYZ,
then ∠A = ∠X, ∠B = ∠Y,
∠C = ∠Z and .ZXCA
YZBC
XYAB
A
B
C
X
Y
Z
In the figure, given that △ABC ~ △PQR,
find the unknowns x, y and z.
y = 98°
The Meaning of Similarity
x = 30° , ∴
BCQR
=CARP
∴5z
=35.4
z = 535.4
= 7.5
In the figure, △ABC ~ △RPQ. Find the values of the
unknowns.
The Meaning of Similarity
Since △ABC ~ △RPQ,
∠B = ∠P
∴ x = 90°
The Meaning of Similarity
Also,RPAB
=PQBC
=y
39
4852
524839
= y
∴ y = 36
Also,RQAC
=PQBC
=60z
4852
486052
z =
∴ z = 65
Back to Question
Three Angles Equal
Conditions for Triangles to be Similar
Example
A)
‧ If two triangles have three pairs of
equal corresponding angles, then
they must be similar.
【 Reference: AAA 】
Show that △ABC and △PQR in the
figure are similar.
Conditions for Triangles to be Similar
In △ABC and △PQR as shown,
∠B = ∠Q, ∠C = ∠R,
∠A = 180° – 35° – 75° = 70°
∠P = 180° – 35° – 75° = 70°
∴ ∠A = ∠P
∴ △ABC ~ △PQR (AAA)
∠sum of
Are the two triangles in the figure similar? Give reasons.
Conditions for Triangles to be Similar
【 In △ABC, ∠B = 180° – 65° – 45° = 70°
In △PQR, ∠R = 180° – 65° – 70° = 45° 】
Yes, △ABC ~ △PQR (AAA).
Three Sides Proportional
Conditions for Triangles to be Similar
Example
B)
‧ If the three pairs of sides of two triangles are
proportional, then the two triangles must be similar.
【 Reference: 3 sides proportional 】
a
b
c d
ef f
ceb
da
Show that △PQR and △LMN in
the figure are similar.
Conditions for Triangles to be Similar
In △PQR and △LMN as shown,
NLRP
164
41
LMPQ
82
41 ,
MNQR
123
41 ,
∴NLRP
MNQR
LMPQ
∴ △PQR ~ △LMN (3 sides proportional)
Are the two triangles in the figure similar? Give reasons.
Conditions for Triangles to be Similar
Yes, △ABC ~ △XZY (3 sides proportional).
【 25.4
9 ZYBC
224
XZAB
, 26
12 XYAC
, 】
Two Sides Proportional and their Included Angle Equal
Conditions for Triangles to be Similar
Example
C)
‧ If two pairs of sides of two triangles are proportional
and their included angles are equal, then the two
triangles are similar.
【 Reference: ratio of 2 sides, inc.∠ 】
x yp
qr
s
yxsq
rp ,
Show that △ABC and △FED in
the figure are similar.
Conditions for Triangles to be Similar
In △ABC and △FED as shown,
∠B = ∠E
FEAB
24 2 ,
EDBC
5.49 2
∴
EDBC
FEAB
∴ △ABC ~ △FED (ratio of 2 sides, inc. )∠
Are the two triangles in the figure similar? Give reasons.
Conditions for Triangles to be Similar
【 ∠ ZYX = 180° – 78° – 40° = 62°, ∠ZYX = ∠CBA = 62°,
236
YZBC
, 224
XYAB
】
Yes, △ABC ~ △XYZ (ratio of 2 sides, inc. ). ∠
Assignment
• Perform Investigation 1 on page 213-214
Homework:
• CYU # 6-11 on pages 215-216.
VECTORS AND BEARINGS
WHAT IS A VECTOR?
It describes the motion of an object.A Vector comprises of
– Direction– Magnitude (Size)
We will consider :– Column Vectors
Column Vectors
a
Vector a
4 RIGHT
2 up
NOTE!
Label is in BOLD.
When handwritten, draw a wavy line under the label
i.e.
~a(4, 2)
Column Vectors
b
Vector b
COLUMN Vector?
3 LEFT
2 up ( -3, 2 )
Column Vectors
n
Vector u
COLUMN Vector?
4 LEFT
2 down( -4, -2 )
Describing Vectors
b
a
c
d
Alternative Labelling
CD00000000000000
EF00000000000000
GH00000000000000
AB
A
B
C
DF
E
G
H
Generalization
Vectors has both LENGTH and DIRECTION.
What is BEARINGS?
It is the angle of direction clockwise from north.
Bearings
P x
Qx
Solution:
Ex. The bearing of R from P is 220 and R is due west of Q. Mark the position of R on the
diagram.
P x
Ex. The bearing of R from P is 220 and R is due west of Q. Mark the position of R on the
diagram.
.
Qx
Solution:
P x
Ex. The bearing of R from P is 220 and R is due west of Q. Mark the position of R on the
diagram.
220.
Solution:
Qx
If you only have a semicircular protractor, you need tosubtract 180 from 220 and measure from south.
P x
Ex. The bearing of R from P is 220 and R is due west of Q. Mark the position of R on the
diagram.
Solution:
Qx
If you only have a semicircular protractor, you need tosubtract 180 from 220 and measure from south.
40.
P x
Ex. The bearing of R from P is 220 and R is due west of Q. Mark the position of R on the
diagram.
220
.
QxR
Solution:
Practice Exercises – Class work
Homework:
1) Research about Pythagorean Theorem and its proof.
2) Answer the following questions:
Check Your Understanding # 14, 15, 16 on pages 218.
Vectors and Bearings