ICSE Class 9 Maths Chapter 10 Triangles

Triangles

Points To Remember

Triangle. A plane figure bounded by three line segments is called a triangle. The line segments forming a triangle are called its sides and each point, where two sides intersect, is called its vertex. We denote a triangle by the symbol △.

Thus, a △ABC has:

(i) three sides, namely AB, BC and CA;

(ii) three vertices, namely A, B and C;

(iii) three angles, namely ∠BAC, ∠ABC and ∠BCA, to be denoted by ∠A, ∠B and ∠C respectively.

A triangle has six elements, namely three sides and three angles.

Exterior Angle of a Triangle. If a side BC of a △ABC is produced to a point D, then ∠ACD is called an exterior angle at C and ∠B and ∠A are called its interior opposite angles.

Types Of Triangles On The Basis Of Sides

(i) Equilateral Triangle. A triangle having all sides equal, is called an equilateral triangle.

In the given figure, in △ABC, we have AB = BC = CA.

(ii) Isosceles Triangle. A triangle having any two sides equal, is called an isosceles triangle.

In the given figure, in △ABC, we have AB = AC.

(iii) Scalene Triangle. A triangle in which all the sides are of different lengths is called a scalene triangle.

In the given figure, in △ABC, we have AB ≠ AC ≠ BC.

Perimeter Of A Triangle

Perimeter of a Triangle. The sum of the lengths of the sides of a triangle is called its perimeter.

Types Of Triangles On The Basis Of Angles

(i) Acute-Angled Triangle. A triangle in which every angle measures more than 0° but less than 90°, is called an acute-angled triangle.

(ii) Right-Angled Triangle. A triangle in which one of the angles measures 90°, is called a right-angled triangle or simply a right triangle.

In a right triangle, the side opposite to the right angle is called its hypotenuse and the other two sides are called its legs.

In △ABC, ∠B = 90°.

It is a right angled triangle in which AC is the hypotenuse and AB, BC are its legs.

(iii) Obtuse-Angled Triangle. A triangle in which one of the angles measures more than 90° but less than 180°, is called an obtuse-angled triangle.

Thus, in an obtuse-angled triangle, one of the angles is obtuse angle.

In △ABC, we have ∠ABC = 120°, which is an obtuse-angle.

△ABC is an obtuse-angled triangle.

Medians Of A Triangle

Medians of a Triangle. The median of a triangle corresponding to any side is the line segment joining the mid-point of that side with the opposite vertex.

In the given figure D, E, F are the mid-points of the sides BC, CA and AB respectively of △ABC.

Thus, AD is the median corresponding to side BC;

BE is the median corresponding to side CA;

CF is the median corresponding to side AB.

A triangle has three medians and the medians of a triangle are concurrent, i.e., they intersect at the same point.

The point of intersection of the medians of a triangle is called its centroid.

In the given figure, G is the centroid of △ABC.

G divides AD in the ratio 2: 1, i.e., AG: GD = 2: 1.

Similarly, BG: GE = 2: 1 and CG: GF = 2: 1.

Altitudes Of A Triangle

Altitudes of a Triangle. The altitude of a triangle corresponding to any side is the length of perpendicular from the opposite vertex to that side.

In the given figure, in △ABC, we have AL ⊥ BC, BM ⊥ CA and CN ⊥ AB.

AL is the altitude corresponding to side BC;

BM is the altitude corresponding to side CA;

CN is the altitude corresponding to side AB.

A triangle has three altitudes and the altitudes of a triangle are concurrent, i.e., they intersect at the same point.

The point of intersection of the altitudes of a triangle is called its orthocentre.

In the given figure, H is the orthocentre of △ABC.

Some Theorems And Their Applications

Theorem 1. The sum of the angles of a triangle is equal to two right angles.

Theorem 2. If one side of a triangle is produced, then the exterior angle so formed is equal to the sum of its interior opposite angles.

Notes. 1. A triangle cannot have more than one right angle or obtuse angle.

2. In a right triangle, sum of two acute angle is 90°.

3. An exterior angle of a triangle is greater than its interior opposite angle.

Congruency Of Triangles

(i) Congruent Figures. Two geometrical figures, having exactly the same shape and size are known as congruent figures.

(ii) Congruent Triangles. Two triangles are congruent if and only if one of them can be made to superpose on the other, so as to cover it exactly.

Thus, congruent triangles are exactly identical.

If △ABC is congruent to △DEF, we write △ABC ≅ △DEF.

This happens when AB = DE, BC = EF, AC = DF and ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.

Criterion For Congruence

(i) (SAS-axiom). If two triangles have two sides and the included angle of the one equal to the corresponding sides and the included angle of the other, then the triangles are congruent.

In the given figure, in △ABC and △DEF, we have:

AB = DE, AC = DF and ∠A = ∠D.

△ABC ≅ △DEF [By SAS-axiom]

(ii) (AAS-axiom). If two triangles have two angles and a side of the one equal to the corresponding two angles and the corresponding side of the other, then the triangles are congruent.

In the given figure, in △ABC and △DEF, we have:

∠A = ∠D, ∠B = ∠E

and BC = EF

△ABC ≅ △DEF [By AAS-axiom]

(iii) (SSS-axiom). If two triangles have three sides of the one equal to the corresponding three sides of the other, then the triangles are congruent.

In the given figure, in △ABC and △DEF, we have:

AB = DE, BC = EF and AC = DF.

△ABC ≅ △DEF [By SSS-axiom]

(iv) (RHS-axiom). In two right-angled triangles if the hypotenuse and one side of the one are equal to the hypotenuse and the corresponding side of the other, then the triangles are congruent.

In the given figure, △ABC and △DEF are right-angled triangles in which Hyp. AC = Hyp. DF and BC = EF.

△ABC ≅ △DEF [By RHS-axiom]

Note. The corresponding parts of two congruent triangles are always equal. We show it by the abbreviation 'c.p.c.t.' which means 'corresponding parts of congruent triangles.'

Isosceles Triangles

Theorem 1. If two sides of a triangle are equal, then the angles opposite to them are also equal.

Theorem 2. (Converse of Theorem 1). If two angles of a triangle are equal, then the sides opposite to them are also equal.

Theorem 3. If two sides of a triangle are unequal, then the greater side has greater angle opposite to it.

Theorem 4. (Converse of Theorem 3). If two angles of a triangle are unequal, then the greater angle has greater side opposite to it.

Theorem 5. The sum of any two sides of a triangle is greater than its third side.

Theorem 6. Of all the line segments that can be drawn to a given straight line from a given point outside it, the perpendicular is the shortest.

Construction Of Triangles

We know that a triangle has six elements, three sides and three angles. Therefore to construct a triangle, atleast three elements are required.

(i) Three sides.

(ii) Two sides and included angle.

(iii) Two angles and included sides. Beside these, we can also construct a triangle with given data as given below.

(iv) Two sides and an altitude to the third side.

(v) To construct an isosceles triangle whose base and height are given.

(vi) To construct an isosceles triangle whose one altitude and vertical angle are given.

(vii) To construct an equilateral triangle whose height is given.

(viii) To construct a right triangle whose one side and hypotenuse is given.

(ix) To construct a triangle whose perimeter and ratio of sides are given.

(x) To construct a triangle whose perimeter and base angles are given.

(xi) To construct a triangle in which base, one base angle and sum of other two sides are given.

(xii) To construct a triangle whose base, one base angle and difference of other two sides are given.

Teacher's Note

Understanding triangles helps us appreciate why triangular shapes are used in bridge construction and roof trusses - they provide maximum strength with minimum material.

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