ICSE Class 8 Maths Chapter 22 Triangles

Chapter 22

Triangles

Triangles

A triangle is a closed figure bounded by three line segments. The adjoining figure shows a triangle ABC. The line segments AB, BC and CA are called its sides and the angles CAB, ABC and BCA are called its interior angles or simply the angles. The points A, B and C are called its vertices. Usually, the triangle ABC is written as \(\triangle ABC\). The three sides and the three angles of a triangle are called its six elements.

Exterior Angles Of A Triangle

Let ABC be a triangle and its sides BC, CA and AB be produced to D, E and F respectively, then \(\angle ACD\), \(\angle EAB\) and \(\angle FBC\) are called exterior angles at C, A and B respectively. The two interior angles of \(\triangle ABC\) that are opposite to the exterior \(\angle ACD\) are called its interior opposite angles. Thus, \(\angle A\) and \(\angle B\) are interior opposite angles of exterior \(\angle ACD\).

Notice that \(\angle BCA\) and \(\angle ACD\) form a linear pair, so \(\angle BCA + \angle ACD = 180°\). Thus, we have:

An exterior angle + adjacent interior angle = 180°

Classification Of Triangles On The Basis Of Sides

(i) Scalene Triangle

If all the three sides of a triangle are unequal, it is called a scalene triangle. In the adjoining figure, AB \(\neq\) BC \(\neq\) CA, so \(\triangle ABC\) is a scalene triangle.

(ii) Isosceles Triangle

If any two sides of a triangle are equal, it is called an isosceles triangle. In the adjoining figure, AB = AC, so \(\triangle ABC\) is an isosceles triangle. Usually, equal sides are indicated by putting marks on each of them.

(iii) Equilateral Triangle

If all the three sides of a triangle are equal, it is called an equilateral triangle. In the adjoining figure, AB = BC = CA, so \(\triangle ABC\) is an equilateral triangle.

Teacher's Note

Triangles are fundamental shapes found everywhere in nature and architecture. From roof trusses in buildings to the structure of molecules, understanding triangles helps us appreciate the design of everyday objects.

Classification Of Triangles On The Basis Of Angles

(i) Acute Angled Triangle

If all the three angles of a triangle are acute (less than 90°), it is called an acute angled triangle. In the adjoining figure, each angle is less than 90°, so \(\triangle ABC\) is an acute angled triangle.

(ii) Right Angled Triangle

If one angle of a triangle is right angle (= 90°), it is called a right angled triangle. In a right angled triangle, the side opposite to right angle is called hypotenuse. In the adjoining figure, \(\angle B = 90°\), so \(\triangle ABC\) is a right angled triangle and side AC is the hypotenuse.

(iii) Obtuse Angled Triangle

If one angle of a triangle is obtuse (greater than 90°), it is called an obtuse angled triangle. In the adjoining figure, \(\angle B\) is obtuse (greater than 90°), so \(\triangle ABC\) is an obtuse angled triangle.

Teacher's Note

Different angle classifications help us identify triangle types at a glance. Carpenters and builders use right-angled triangles extensively when checking if corners are truly square.

Some Terms Connected With A Triangle

Altitude

Perpendicular from a vertex of a triangle to the opposite side is called an altitude of the triangle. In the adjoining figure, AD \(\perp\) BC, so AD is an altitude to the side BC. A triangle has three altitudes. In fact, all the altitudes pass through the same point and the point of concurrence is called the orthocentre of the triangle. In the adjoining figure, AD, BE and CF are the altitudes of \(\triangle ABC\) and the point H is the orthocentre of \(\triangle ABC\).

Median

Line joining a vertex of a triangle to the mid-point of the opposite side is called a median of the triangle. In the above figure, D is mid-point of BC, so AD is a median of \(\triangle ABC\). A triangle has three medians. In fact, all the medians pass through the same point and the point of concurrence is called the centroid of the triangle. In the above figure, AD, BE and CF are the medians of \(\triangle ABC\) and the point G is the centroid of \(\triangle ABC\).

Teacher's Note

Altitudes and medians are construction tools that help us understand triangle properties. The centroid of a triangle is its balance point - which is why it's important in engineering and design.

Incentre And Incircle

Line bisecting an (interior) angle of a triangle is called the (internal) bisector of the angle of the triangle. In the adjoining figure, \(\angle BAI = \angle IAC\), so AI is the (internal) bisector of \(\angle A\). A triangle has three internal bisectors of its angles. In fact, all the (internal) bisectors of the angles of a triangle pass through the same point and the point of concurrence is called the incentre of the triangle. In the above figure, IA, IB and IC are the (internal) bisectors of \(\angle A\), \(\angle B\) and \(\angle C\) respectively of \(\triangle ABC\). So I is the incentre of \(\triangle ABC\). Moreover, incentre is the centre of a circle which touches all the sides of \(\triangle ABC\) and this circle is called incircle of \(\triangle ABC\).

Circumcentre And Circumcircle

Line bisecting a side of a triangle and perpendicular to it is called the right bisector of the side of the triangle. In the adjoining figure, D is mid-point of BC and OD \(\perp\) BC, so OD is the right bisector of the side BC. A triangle has three right bisectors of its sides. In fact, all right bisectors of the sides of a triangle pass through the same point and the point of concurrence is called the circumcentre of the triangle. In the above figure, OD, OE and OF are the right bisectors of the sides of \(\triangle ABC\). So O is the circumcentre of \(\triangle ABC\). Moreover, circumcentre is the centre of a circle which passes through the vertices of \(\triangle ABC\) and this circle is called circumcircle of \(\triangle ABC\).

Teacher's Note

The incentre and circumcentre are special points with practical applications. In city planning, the circumcentre helps determine optimal placement of emergency services to serve all areas equally.

Angles Property Of A Triangle

Theorem

The sum of angles of a triangle is 180°.

Given. A triangle ABC.

To prove. \(\angle A + \angle B + \angle C = 180°\)

Construction. Produce BC to D and through C draw a line CE parallel to BA.

Proof.

Q.E.D.

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