GEOMETRY(TRIANGLES)GEOMETRY

(TRIANGLES)

MAIN MENUOBJECTIVESOBJECTIVES

INTRODUCTIONINTRODUCTION

CHARACTERISTIC OF TRIANGLES

CHARACTERISTIC OF TRIANGLES

TYPES OF TRIANGLETYPES OF TRIANGLE

CONSTRUCTING TRIANGLES

CONSTRUCTING TRIANGLES

ACTIVITIESACTIVITIES

OBJECTIVES

Identify the types of triangles correctly

Identify the characteristics of triangles clearly

Construct a triangles using method correctly

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INTRODUCTION

A triangle is a geometrical figure formed by three lines, which intersect each other and which are not all concurrent.

A triangle ABC has six parts namely three angles and their

sides namely AB,BC, CA.

Triangles can be classified either on the basis of the sides or on the basis of the angles.

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CHARACTERISTIC OF TRIANGLES

θ°

λ°

Height

ApexApex angle

Side

Base angle

Base

Bisection line

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TYPES OF TRIANGLE

a) Equilateral triangle

b) Right triangle

An equilateral triangle has got 3 sides of equal length and 3 angles that are equal.

Since ALL the angles in a triangle add up to 180º then 180 divided by 3 must be 60º.

The right angled triangle contains a right angle (an angle of 90˚).

TYPES OF TRIANGLE

d) Scalene triangle

c) Isosceles triangle

An Isosceles triangle has got two sides of equal length and 2 angles equal.

A scalene triangle is the easiest of them all. The scalene triangle has got NO sides of equal length and NO angles the same.

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CONSTRUCTING TRIANGLES1.CONSTRUCTING A TRIANGLE GIVEN THREE SIDES1.CONSTRUCTING A TRIANGLE GIVEN THREE SIDES

2.CONSTRUCTING A TRIANGLE GIVEN TWO SIDES AND AN ANGLE2.CONSTRUCTING A TRIANGLE GIVEN TWO SIDES AND AN ANGLE

3.CONSTRUCTING A TRIANGLE GIVEN TWO SIDES AND A BASE ANGLE3.CONSTRUCTING A TRIANGLE GIVEN TWO SIDES AND A BASE ANGLE

4.CONSTRUCTING A TRIANGLE GIVEN ITS BASE AND TWO BASE ANGLES4.CONSTRUCTING A TRIANGLE GIVEN ITS BASE AND TWO BASE ANGLES

5.CONSTRUCTING A TRIANGLE GIVEN THE PERIMETER AND SIDE RATIOS5.CONSTRUCTING A TRIANGLE GIVEN THE PERIMETER AND SIDE RATIOS

6.CONSTRUCTING A RIGHT TRIANGLE GIVEN THE HYPOTENUSE

AND LENGTH OF A SIDE6.CONSTRUCTING A RIGHT TRIANGLE GIVEN THE HYPOTENUSE

AND LENGTH OF A SIDE

7.CONSTRUCTING A TRIANGLE GIVEN THE BASE,

BASE ANGLE AND APEX ANGLE7.CONSTRUCTING A TRIANGLE GIVEN THE BASE,

BASE ANGLE AND APEX ANGLE

8.CONSTRUCTING A TRIANGLE GIVEN THE BASE, HEIGHT AND APEX ANGLE8.CONSTRUCTING A TRIANGLE GIVEN THE BASE, HEIGHT AND APEX ANGLE

9.CONSTRUCTING A TRIANGLE GIVEN THE PERIMETER, HEIGHT AND APEX ANGLE9.CONSTRUCTING A TRIANGLE GIVEN THE PERIMETER, HEIGHT AND APEX ANGLE

10.CONSTRUCTING A TRIANGLE GIVEN THREE BISECTOR LINES10.CONSTRUCTING A TRIANGLE GIVEN THREE BISECTOR LINES MENU

PQ

a

babc

c

R

1. CONSTRUCTING A TRIANGLE GIVEN THREE SIDES

Given the sides a, b and c

STEP 1 : Draw the side a. On the point P, draw an arc with side b.

STEP 2 : Construct an arc c from QSTEP 3 : Draw the lines PR and QR

2. CONSTRUCTING A TRIANGLE GIVEN TWO SIDES AND AN ANGLE

a b

aθ0 θ0

QP

C

b

Given two sides a and b and an included angle θ0 STEP 1 : Draw sides a and construct the

angle 00STEP 2 : Construct an arc with side b from PSTEP 3 : Draw the lines PC and QC

3. CONSTRUCTING A TRIANGLE GIVEN TWO SIDES AND A BASE ANGLE

Given two sides a, b and an angle θ0

θ0

QP

θ0a

R’

Rb

STEP 1 : Draw sides a and construct the angle 00

STEP 2 : Construct an arc with side b from Q intersecting at R and R’

STEP 3 : Draw the lines PR and QR. The triangle PQR’ is a alternative to triangle PQR

a

b

4. CONSTRUCTING A TRIANGLE GIVEN ITS BASE AND TWO BASE ANGLES

β0α0

a

α0

QP

a

R

β0

STEP 2 : Construct angle β0 as to intersect at RSTEP 1 : Draw side a and construct angle α0

STEP 3 : Draw the lines PR and QR

Given the base a and base angles α0 and β0

5. CONSTRUCTING A TRIANGLE GIVEN THE PERIMETER AND SIDE RATIOS

Q

P

4

6

3

STEP 3 : Draw the lines CD, CE and DESTEP 2 : Centred at C and D, construct arcs with radii PC and DQ intersecting at ESTEP 1 : Divide the line PQ into 4:6:3 ratio. Mark C and D

P

Q

C

D

E

Given perimeter PQ

6. CONSTRUCTING A RIGHT TRIANGLE GIVEN THE HYPOTENUSE AND LENGTH OF

A SIDE

Hypotenuse, h

a

P Q

R

Given hypotenuse and side a

h

STEP 1 : Draw and bisect the hypotenuse. Construct a semicircleSTEP 2 : Centred at P, construct an arc with radius a intersecting at RSTEP 3 : Draw the lines PR and QR

7. CONSTRUCTING A TRIANGLE GIVEN THE BASE, BASE ANGLE AND APEX ANGLE

a

θ0β0

Given the base a, base angles θ 0

and apex angle β0.

β0

A θ0 B

STEP 1 : Draw the base a and construct the apex angle θ. Construct a perpendicular line at A. Construct the angle β 0 at B.

STEP 2 : Bisect AB so as to intersect at O. Centered at O, construct a circle through A and B so as to intersect at C.

STEP 3 : Draw the lines AC and BC

C

8. CONSTRUCTING A TRIANGLE GIVEN THE BASE, HEIGHT AND APEX ANGLE

h

a

θ0

h

a

BA

STEP 1 : Draw base a and construct the apex angle θ 0 . Construct a perpendicular line at A. Construct a parallel line h from AB.

Given the height h, base a and apex angle θ0 .

a

BA

O

C C’

STEP 2 : Bisect AB so as to intersect at O. Centered at O, construct a circle through A and B so as to intersect at C.

STEP 3 : Draw the lines AC and BC.

θ0

BA

C C’

h

9. CONSTRUCTING A TRIANGLE GIVEN PERIMETER, HEIGHT AND APEX ANGLE

hpθ0

Given p, height h and

apex angle θ0.

θ0

O

A

QP

R

h+OPN

B

h

C

STEP 1: Draw ½ perimeter AP and AQ and the apex angle θ0 .STEP 2 : Centered at O, construct an arc through P and Q.STEP 3: Draw AO and construct a half circle with diameter AO.STEP 4: Centered at O, construct (h+OP) intersecting at M and N.STEP 5: Draw perpendicular lines at N intersecting at B and C. Construct triangle ABC. AB+BC+AC = p.

10. CONSTRUCTING A TRIANGLE GIVEN THREE BISECTOR LINES

A a

B b

C c

Given the three bisector lines of a triangle.

2/3 1/3

A a

1

2

3

h

STEP 1: Draw middle line Aa and divide into three parts.STEP 2 : Centered at a, construct an arc with radius 1/3 Aa intersecting at x.STEP 3: Centered at x, construct arc with radius 2/3 Bb and at h, with radius 2/3 Cc intersecting at C.

STEP 4: Centered at h, construct an arc with radius 2/3 Bb. Extend line Ca to intersect at B.

STEP 5: Draw the triangle ABC.

x

B b C c

2/3 2/3

2/3 Cc 2/3 Bb

C

B

A ah

2/3 Bb

ACTIVITIES

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ACTIVITY 1ACTIVITY 1

ACTIVITY 2ACTIVITY 2

ACTIVITY 1

Right triangle Isosceles triangle

Right triangle

Can you find the triangles in this picture?

Equilateral triangle Equilateral

triangle

ACTIVITY 2

Right triangle

Isosceles triangle

Equilateral triangle

Scalene triangle

Right triangle

Isosceles triangle

Equilateral triangle

Scalene triangle

Right triangle

Isosceles triangle

Equilateral triangle

Scalene triangle

Right triangle

Isosceles triangle

Equilateral triangle

Scalene triangle

Right triangle

Isosceles triangle

Equilateral triangle

Scalene triangle