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A triangle is called an isosceles triangle if it has any two sides equal. The angles opposite to these equal sides are also equal. The area of an isosceles triangle can be calculated using the length of its sides. In the diagram, triangle ABC, here sides AB and AC are equal and also ∠B = ∠C. The theorem that describes the isosceles triangle is “if the two sides of a triangle are congruent, then the angle opposite to these sides are congruent”.

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As we know the perimeter of any shape is given by the boundary of the shape. In a similar way, the perimeter of an isosceles triangle is defined as the sum of the three sides of an isosceles triangle. The perimeter of an isosceles triangle can be found if we know its base and side. The formula of isosceles triangle perimeter is given by:

The perimeter of an isosceles triangle formula, P = 2a + b units

where ‘a’ is the length of the two equal sides of an isosceles triangle and ‘b’ is the base of the triangle.

The area of an isosceles triangle is defined as the region occupied by it in the two-dimensional space. Generally, the isosceles triangle is half the product of the base and height of an isosceles triangle. Formula to calculate the area of an isosceles triangle is given below:

The area of an isosceles triangle A = ½ × b × h Square units, where ‘b’ is the base and ‘h’ is the height of the isosceles triangle.

As we know the two sides are equal in this triangle, and the unequal side is called the base of the triangle.

The angles opposite to the two equal sides of the triangle are always equal.

The height of an isosceles triangle is measured from the base to the vertex (topmost) of the triangle.

The third angle of a right isosceles triangle is equal to 90 degrees.

Generally, the isosceles triangle is classified into different types named as,

Isosceles acute triangle.

Isosceles right triangle.

Isosceles obtuse triangle.

Let’s discuss in detail these three different types of an isosceles triangle.

As we know that the different dimensions of a triangle are legs, base, and height. The axis of symmetry of an isosceles triangle is along the perpendicular bisector of its base. Depending upon the angle between the two legs, the isosceles triangle is classified as acute, right, and obtuse. The isosceles triangle can be acute if the two angles opposite to the legs are equal and are less than 90 degrees ( or acute angle).

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A right isosceles triangle has two equal sides, in which one of the two equal sides act as perpendicular and another one as a base of the triangle. The third side, which is unequal, is known as the hypotenuse. Therefore, we can apply here the Pythagoras theorem, where the square of the hypotenuse is equal to the sum of the square of base and perpendicular.

Suppose, the sides of the right isosceles triangle are a, a, and h, where a is the length of two equal sides and h is the length of the hypotenuse, then;

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h = √(a^{2} + a^{2}) = √2a^{2} = a√2

or h = √2 a

We know that the obtuse triangle is a triangle in which one of its angles is greater than 90 degrees (or right angle). Also, it is not possible to draw a triangle with more than two obtuse angles. We know that the obtuse triangle can be two types, i.e., scalene triangle or isosceles triangle. Therefore, the isosceles obtuse triangle is a triangle, which has two equal sides having an obtuse angle.

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1. Find the perimeter of an isosceles triangle, with a side of 5 cm and a base of 4 cm.

Solution: Given that, length of the base is 4 cm.

The length of the two equal sides is 5 cm.

We know that the formula to calculate the perimeter of an isosceles triangle is P = 2a + b units.

Now, substitute the value of base and side in the perimeter formula, we get

P = 2(5) + 4 = 10 + 4 = 14 cm

Therefore, the perimeter of an isosceles triangle is 16 cm.

2. Find the area of an isosceles triangle given its height as 5 cm and base as 4 cm?

Solution: Given that, base = 4 cm and height = 5 cm

We know that the area of an isosceles triangle is ½ × b × h square units

Now, substitute the base and height value in the formula to calculate the area

Area of an isosceles triangle is ½ × b × h

A = ½ × 4 × 5 = 10 cm^{2}

Therefore, the area of an isosceles triangle is 10 cm^{2}

An isosceles triangle is a triangle that has any of its two sides of the same length, and also two angles opposite to the equal sides will be of equal measure. The perimeter and area of an isosceles triangle are calculated by using the lengths of its equal sides and of its base. There are various applications of isosceles triangles in mathematics and constructions. Hence it is one of the most fundamental concepts of geometry.

FAQ (Frequently Asked Questions)

1. Can the Isosceles Triangle has an Obtuse Angle?

Ans: Yes, an isosceles triangle can have only one obtuse angle. An angle whose measure is greater than 90 degrees and lesser than 180 degrees is known as an obtuse angle.

2. What is the Measure of the Two Equal Angles of a Right Isosceles Triangle?

Ans: As we know in an isosceles right triangle two sides are congruent and the corresponding angles will be 45 degrees each which sums to 90 degrees. Hence the measure of the two equal angles of an isosceles right triangle is 45 degrees each.