Of

Triangle

Properties

REMYA S

13003014

MATHEMATICS

MTTC PATHANAPURAM

The Triangle and its Properties

Triangle is a simple closed curve made of three line

segments.

Triangle has three vertices, three sides and three angles.

In Δ ABC

Sides: AB, BC and CA

Angles: ∠BAC, ∠ABC and ∠BCA

Vertices: A, B and C

The side opposite to the vertex A is BC.

Based on the sides

Scalene Triangles

No equal sides

No equal angles

Isosceles Triangles

Two equal sides

Two equal angles

Equilateral Triangles

Three equal sides

Three equal angles,

always 60°

Classification of triangles

Scalene

Isosceles

Equilateral

Classification of triangles Based on Angles

Acute-angled Triangle

All angles are less than 90°

Obtuse-angled Triangle

Has an angle more than 90°

Right-angled triangles

Has a right angle (90°)

Acute

Triangle

Right

Triangle

Obtuse

Triangle

MEDIANS OF A TRIANGLE A median of a triangle is a line segment joining

a vertex to the midpoint of the opposite side

A triangle has three medians.

• The three medians always meet at a single point.

• Each median divides the triangle into two smaller

triangles which have the same area

• The centroid (point where they meet) is the center of gravity of

the triangle

.

ALTITUDES OF A TRIANGLE• Altitude – line segment from a vertex

that intersects the opposite side at a right angle.

Any triangle has three altitudes.

Definition of an Altitude of a Triangle

A segment is an altitude of a triangle if and only if ithas one endpoint at a vertex of a triangle and theother on the line that contains the side opposite thatvertex so that the segment is perpendicular to this line.

ACUTE OBTUSE

B

A

C

ALTITUDES OF A TRIANGLE

RIGHT

A

B C

If ABC is a right triangle, identify its altitudes.

BG, AB and BC are its altitudes.

G

Can a side of a triangle be its altitude? YES!

ALTITUDES OF A TRIANGLE

Proof: C + D + E = 1800 ……..Straight line

A = D and B = E….Alternate angles

C + B + A = 1800

A + B + C = 1800

D E

Given: Triangle

A B

C

Construction: Draw line ‘l’ through C parallel

to the base AB

The measure of the three angles of a triangle sum

to 1800 .

To Prove : A + B + C = 1800

l

ANGLE SUM PROPERTY OF A

TRIANGLE

An exterior angle of a triangle equals the sum of the

two interior opposite angles in measure.

To Prove: ACD = ABC + BAC

Proof: CB + ACD = 1800 …………………. Straight line

ABC + ACB + BAC = 1800 …………………sum of the triangle

ACB + ACD = ABC + ACB + BAC

ACD = ABC + BAC

A

B C D

Given: In Δ ABC extend BC

to D

EXTERIOR ANGLE OF A TRIANGLE

AND ITS PROPERTY

PYTHAGORAS THEOREMIn a right angled triangle the square of the hypotenuse is

equal to the sum of the squares of the other two sides.

In ABC :

• AC is the hypotenuse

• AB and BC are the 2 sides

Then according to Pythagoras theorem ,

A

B C

AC² = AB² + BC²