Isosceles Triangle

What is a Triangle?

A triangle is a closed 2D figure having three sides, three vertices and three angles. It is the simplest form of Polygon. A triangle can be formed by joining any three dots such that the line segments connect each other end to end. Three line segments connecting the dots are the sides of the triangle, the point of intersection of two lines is known as vertex and the space between them is called an angle. It is also important to know that the sum of all the interior angles of a triangle is always 180 degrees.

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Triangles are the simplest form of Polygon. The meaning of the word Polygon is a plane figure which has many line segments connected end to end. A single or double line segments together can never form a polygon. The line joining any three dots can form a triangle or any three line segments connected to each other end to end also forms a triangle. A triangle is a three-sided closed 2D shape having three vertices as well as three angles. 

Types Of Triangles:

Triangles can be classified on the basis of their size as well as angles.

Classification of Triangles on the Basis of their Sides is as Follows:

  1. Equilateral Triangle: An equilateral triangle is one whose all the three sides are equal.

  2. Isosceles Triangle: An isosceles triangle is one whose two sides are equal.

  3. Scalene Triangle:  A scalene triangle is one whose all three sides are unequal.

Classification of Triangles on the Basis of their Angles is as Follows:

  1. Acute Angled Triangle: A triangle whose all interior angles are less than 900.

  2. Right Angled Triangle: A triangle whose one interior angle is 900.

  3. Obtuse Angled Triangle: A triangle whose one interior angle is more than 900.

What Is the Isosceles Triangle?

 A triangle having two sides of equal length is called the Isosceles triangle. In an isosceles triangle, the angles that are opposite to the equal sides are equal. In the triangle given below, two sides are of 5inches and one side is 3 inches. Thus, it is an Isosceles Triangle.

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Properties of Isosceles Triangle:

  1. Two sides of an Isosceles triangle are congruent to each other.

                     

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In the above figure, sides AB and AC are of equal length ‘a’ unit.

  1. The unequal side of an isosceles triangle is called a base.

  2. The two angles opposite to the equal sides are congruent to each other. Thus it has two congruent base angles. ∠ B and ∠C are of equal measure.

  3. Apex angle is the angle that is not congruent to the two base angles which are congruent.

  4. The height drawn from the apex of an isosceles triangle divides the base into two equal parts and also divides the apex angle into two equal angles.

  5. Area of Isosceles triangle = ½ × base × height

  6. The perimeter of an Isosceles triangle = sum of all the three sides

  7. The third unequal angle of an isosceles can be acute or obtuse.

  8. The circumcenter of an isosceles triangle lies inside the triangle if all the three angles of the three triangles are acute. 

  9. The sides of the triangle are the chords of the circumcircle.

  10. If one of the angles is 90 degrees, then the circumcenter lies outside the triangle. 

  11. The centroid is the intersection of the medians of the Isosceles triangle. 

  12. The median which is drawn from Apex divides the triangle at right angles.

  13. The perpendicular bisectors of an isosceles triangle intersect at its circumcenter.

  14. The angle bisectors of an isosceles triangle intersect at the incenter.

  15. The circle that is drawn with the incenter touches the three sides of the triangle internally.

  16. Each median divides the isosceles triangle into two equal triangles having the same area.

  17. The area of the triangle can be estimated:

  1. If the measure of one angle and one side are given

  2.  If three sides of the triangle are given.

  3.  If two sides of an isosceles triangle and their included angles are given.

  1. By joining the midpoint of three sides divides the triangle into 4 smaller triangles of the same area.

  2. When a circle is drawn with the diameter equal to the base:

  1. For an obtuse-angled isosceles triangle, the apex lies inside the circle. 

  2. In a right-angled isosceles triangle, the apex lies on the circumference. 

  3. In an acute-angled isosceles triangle, the apex lies outside the triangle.

  1. When the midpoint apex is taken as a radius and a circle is drawn with the midpoint of the base as the centre.

  1. For an acute-angled isosceles, the base vertices lie inside the circle.

  2. For a right-angled isosceles the base vertices lie on the circumference

  3. For an obtuse-angled isosceles triangle, the base vertices lie outside the circle.

In the right-angled isosceles triangle, the altitude on the hypotenuse is always half the length of the hypotenuse.

In the right-angled isosceles triangle, the centre of the circumcircle lies on the hypotenuse and the radius of the circumcircle is half the length of the hypotenuse.

An isosceles triangle is whose two equal sides length is ‘a’ unit and length of its base is ’b’ unit. 

    Area of an Isosceles Triangle 

It is a triangle having two sides of equal length. The area of an isosceles triangle of  two equal sides a, base as b and height as h is \[\frac{{hb}}{2}\]

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             =  \[\frac{b}{2}\sqrt {{a^2} - \frac{{{b^2}}}{4}} \]

Perimeter of a Triangle?

The perimeter of a triangle is the sum of the length of its 3 sides.

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Perimeter of triangle (P) = Side1 + Side2  +Side3

Here a, b and c are the 3 sides of the triangle.

Thus, the perimeter of the triangle (P) = a+b+c

The perimeter of an isosceles triangle 

Two sides of the isosceles triangle are equal.

So, the perimeter of an Isosceles triangle = 2a + c

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Example: In triangle ABC, 2 sides of the triangle (a) = 20 cm

Base (c) = 8 cm

Perimeter = 2(20) + 8

      = 48 cm.

FAQ (Frequently Asked Questions)

Question: What are Some of the Real Life Examples of an Isosceles Triangle?

Solution: The application of an isosceles triangle can be done in real life too.

For example: 

i) To make the roof of a house 

ii) The steeple on a church 

iii) To make an arrowhead