अध्याय 6: त्रिभुज और उसके गुण Class 7 Math

Introduction

Triangle

• A triangle is a closed curve made of three line segments.

• It has three:Sides:
  (i) Sides: ¯¯¯¯¯¯¯¯AB, ¯¯¯¯¯¯¯¯BC and ¯¯¯¯¯¯¯¯CA
  (ii) Angles: ∠BAC, ∠ACB and ∠CBA
  (iii) Vertices: A, B and C

Important Lines in a Triangle

Median

• Median is the line that connects a vertex of a triangle to the mid-point of the opposite side.
• In the given figure, ¯¯¯¯¯¯¯¯¯AD is the median, joining the vertex A to the midpoint of ¯¯¯¯¯¯¯¯BC.

Altitude

• An altitude is a line segment through a vertex of the triangle and perpendicular to a line containing the opposite side.

Sides Also Have Constraints

Sum of the lengths of two sides of a triangle

• The sum of the lengths of any two sides of a triangle is greater than the third side.

• In the above triangle,
  9+11=20 >14
  11+14=25 >9
  9+14=23 >11

Difference between lengths of two sides of a triangle

• The difference between lengths of any two sides is smaller than the length of the third side.

• In the above triangle,
  11 – 9 = 2 < 14
  14 – 11 = 3 < 9
  14 – 9 = 5 < 11
• Using the concept of sum of two sides and difference of two sides, it is possible to determine the range of lengths that the third side can take.

Triangle Properties

Angle sum property of a triangle

• The total measure of the three angles of a triangle is 180⁰.

• In △PQR,
  ∠RPQ+∠PQR+∠QRP
  =70⁰ + 60⁰ + 50⁰ = 180⁰

Exterior angle of a triangle and its property

• An exterior angle of a triangle is equal to the sum of its interior opposite angles.

In the given figure, ∠1+∠2=∠3.

Pythagoras Theorem

• The side opposite to the right angle in a right-angled triangle is called the hypotenuse.
• The other two sides are known as legs of the right-angled triangle.
• In a right-angled triangle, square of hypotenuse is equal to the sum of squares of legs.

AC²=AB²+BC²
⇒5²=4²+3²

Properties of isosceles and equilateral triangles

Properties of Isosceles Triangle 

• Two sides are equal in length.
• Base angles opposite to the equal sides are equal.

Properties of Equilateral Triangle

• All three sides are equal in length.
• Each angle equals to 60⁰.

Classification of Triangles

Classification of triangles based on sides

• Equilateral triangle: A triangle in which all the three sides are of equal lengths.
• Isosceles triangle: A triangle in which two sides are of equal lengths.
• Scalene Triangle: A triangle in which all three sides are of different length.

Classification of triangles based on angles

• Acute-angled: A triangle with three acute angles.
• Right-angled: A triangle with one right angle.
• Obtuse-angled: A triangle with one obtuse angle.