Math 10 Ms. Albarico. A.Modeling Situations Involving Right Triangles B.Congruence and Similarity...

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.


• 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...


• 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


Side AC is across from our reference angle B. So it is labeled: opposite.

A

B (ref. angle)

Copposite

hypotenuse


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


• 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


• Given the same triangle, how would the sides be labeled if angle C were the reference angle?

• Will there be any difference?


• 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


However, side AB is now the side opposite since it is across from angle C.

A

B

C (ref. angle)

oppositehypotenuse


That leaves side AC to be labeled as the adjacent side.

A

B

C (ref. angle)adjacent

hypotenuseopposite


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.


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.


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.


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.


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


(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


Translation

Yes

Enlargement

No

Back to Question


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.


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


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.


(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?


18

10

18

10A B

3

7

537

5

B

Two Sides and Their Included Angle Equal


Example

B)

‧ If AB = XY, ∠B = ∠Y and BC = YZ,

then △ABC △XYZ.

【 Reference: SAS 】

A

B C

X

Y Z



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.


(a) (b)

(a) △ABC △CDA (SSS)

(b) △ACB △ECD (SAS)

Two Angles and One Side Equal


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


Example

C)

2. If ∠A = ∠X, ∠B = ∠Y and BC = YZ,

then △ABC △XYZ.

【 Reference: AAS 】

C

A B

Z

X Y



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.


△ABD △ACD (ASA)

Fulfill Exercise Objective

Identify congruent triangles from given diagram and given reasons.


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.


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.


Yes, △ABD △CDB (RHS)

Similar Figures


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


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.


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°


x = 30° , ∴

BCQR

=CARP

∴5z

=35.4

z = 535.4

= 7.5

In the figure, △ABC ~ △RPQ. Find the values of the

unknowns.


Since △ABC ~ △RPQ,

∠B = ∠P

∴ x = 90°


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


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.


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.


【 In △ABC, ∠B = 180° – 65° – 45° = 70°

In △PQR, ∠R = 180° – 65° – 70° = 45° 】

Yes, △ABC ~ △PQR (AAA).

Three Sides Proportional


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.


In △PQR and △LMN as shown,

NLRP

164

41

LMPQ

82

41 ,

MNQR

123

41 ,

∴NLRP

MNQR

LMPQ

∴ △PQR ~ △LMN (3 sides proportional)



Yes, △ABC ~ △XZY (3 sides proportional).

【 25.4

9 ZYBC

224

XZAB

, 26

12 XYAC

, 】

Two Sides Proportional and their Included Angle Equal


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.


In △ABC and △FED as shown,

∠B = ∠E

FEAB

24 2 ,

EDBC

5.49 2

∴

EDBC

FEAB

∴ △ABC ~ △FED (ratio of 2 sides, inc. )∠



【 ∠ 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


diagram.

.

Qx

Solution:

P x


diagram.

220.

Solution:

Qx

If you only have a semicircular protractor, you need tosubtract 180 from 220 and measure from south.

P x


diagram.

Solution:

Qx

If you only have a semicircular protractor, you need tosubtract 180 from 220 and measure from south.

40.

P x


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

Math 10 Ms. Albarico. A.Modeling Situations Involving Right Triangles B.Congruence and Similarity...

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