**Properties of Isosceles Triangle**

According to the length of their sides, triangles in the world of polygons can be divided into three types: equilateral, isosceles, and scalene. Particularly isosceles triangles have two sides that are of equal length. The goal of this article is to examine the concept, traits, and numerous isosceles triangle instances.

**What is an Isosceles Triangle?**

Let’s do a quick task to better understand the idea of isosceles triangles. A triangle with two equal sides is known as an isosceles triangle. Take a rectangular piece of paper, fold it in half, and then draw a line connecting the top folded corner to the bottom edge to represent this. When you unfold the sheet, you will see that the line has created a triangle. O, D, and C should be written as the triangle’s vertices. Next, determine how long OD and OC are. Observe the pattern by performing this task multiple times with different measurements. That OD and OC are consistently equal will become clear. An isosceles triangle is one of a certain type that has two sides that are the same length.

**Isosceles Triangle Definition**

A triangle with two or more equal-length sides is known as an isosceles triangle. The triangle is also referred to as an equilateral triangle if its three sides are all equal. Angles that are unknown can often be determined using isosceles triangles.

**Properties of Isosceles Triangle**

Geometric shapes have distinctive and special qualities. Isosceles triangles are characterised by the following characteristics:

1.Isosceles triangles have two sides and two angles that are both equal.

2.An isosceles triangle has two equal sides, known as the legs, and an angle between them, known as the vertex angle or apex angle.

3.The side opposite the vertex angle is the base of an isosceles triangle, and the base angles are equal.

4.The base and apex angles are divided by the perpendicular taken from the apex angle.

5.The isosceles triangle is divided into two congruent triangles by the perpendicular from the apex angle, which also serves as the triangle’s symmetry line.

**Isosceles Triangle Angles**

Like other triangles, an isosceles triangle has three angles that add up to 180 degrees. The isosceles triangle theorem states that the angles in an isosceles triangle that face equal sides are also equal. As a result, we have B = C in the isosceles triangle ABC where AB = AC.

The apex angle will be obtuse if the sum of the equal angles is less than 45°. The apex angle will be a right angle when each of the equal angles is precisely 45°. The apex angle, on the other hand, will be acute if each of the equal angles has a measurement greater than 45° but less than 90°.

**Scalene Equilateral and Isosceles Triangle**

Triangles can be divided into three primary categories: scalene, equilateral, and isosceles triangles. Each category has certain characteristics that set it apart from the others. Three sides and three angles with various lengths make up a scalene triangle. An isosceles triangle has two equal sides and two equal angles, whereas an equilateral triangle has three equal sides and three equal angles.

**Types of Isosceles Triangle**

Typically, the isosceles triangle is divided into several categories, including

acute isosceles triangle

Right isosceles triangle

Triangle with isosceles angles

**Isosceles Acute Triangle**

A triangle having two equal sides and two equal angles is known as an isosceles triangle. A triangle’s legs, base, and height are commonly used to calculate its dimensions. The perpendicular bisector of the base forms the axis of symmetry for any isosceles triangle. An isosceles triangle can be right, acute, or obtuse depending on the angle created by the two legs. The isosceles triangle is referred to as acute when the two angles across from the legs are equal and less than 90 degrees.

**Isosceles Right Triangle**

Two of the sides of an isosceles triangle, which is also a right triangle, are equal in length, one of them serving as the perpendicular and the other as the base. The hypotenuse is the third side of the triangle, which is not the same length as the other two sides. The Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the base and perpendicular sides, is applicable in this situation.

If the two equal sides of the right isosceles triangle are marked by the letters a, a, and h, and the hypotenuse is represented by the letter h.

**h = √(a ^{2} + a^{2}) = √2a^{2} = a√2or h = √2 a**

**Isosceles Obtuse Triangle**

We are aware that an obtuse triangle is one with a right angle that is bigger than 90 degrees. Additionally, it is impossible to draw a triangle with more than two obtuse angles. We are aware that an isosceles or scalene triangle can be an obtuse triangle. A triangle having two equal sides and an obtuse angle is called an isosceles obtuse triangle.

**Isosceles Triangle Formulas**

We are aware that an isosceles triangle is a three-sided, two-dimensional shape. The area and perimeter are the measurements used to calculate the isosceles triangle. Let’s now go into more detail on the isosceles triangle’s area and perimeter.

**Area of Isosceles Triangle**

The amount of space an isosceles triangle takes up in two dimensions is referred to as its area. An isosceles triangle’s area is typically equal to half the sum of its base and height. Use the following formula to determine an isosceles triangle’s area:

**The area of an isosceles triangle A = ½ × b × h Square units**

**Perimeter of Isosceles Triangle**

As is well known, a shape’s perimeter serves as its boundary. The lengths of all three sides of an isosceles triangle can also be added up to determine its perimeter. An isosceles triangle’s perimeter can be determined using the following formula if its base and one side are known:

**Perimeter of an isosceles Triangle, P = 2a + b units**

**Isosceles Triangle Altitude**

An isosceles triangle’s vertex angle is split in half when its base is drawn with an altitude, and two congruent triangles are produced as a result. The leg that both triangles share is the triangle’s altitude, which makes a right angle. The hypotenuses of the two freshly produced triangles are also congruent with the congruent legs of the initial triangle. Thus, it can be said that an isosceles triangle’s base is divided by the altitude drawn to it.

**Solved Examples**

**Question 1**

**With a base of 4 cm and a height of 6 cm, what is the area of an isosceles triangle?**

**Solution:**

Due to that,

Height is 6 cm, with a base of 4 cm.

We are aware that an isosceles triangle has an area of 1/2 b h square units.

Replace the base and height values in the formula now.

An isosceles triangle has a surface area of 1/2 b h.

A = ½ × 4 × 6 = 12 cm2

As a result, an isosceles triangle has an area of 12 cm2.

**Question 2**

**The perimeter of an isosceles triangle with a side of 6 cm and a base of 4 cm can be found.**

**Solution:**

Base is equal to 4 cm.

The two equal arms measure 6 cm in length.

We are aware that P = 2a + b units is the formula to determine an isosceles triangle’s perimeter.

Using the perimeter formula now with the values substituted, we obtain

P = 2(6) + 4 = 12 + 4 = 16 cm

Consequently, an isosceles triangle has a 16 cm perimeter. An isosceles triangle’s area with a height of 6 cm and a base of.

**Frequently Asked Questions on Isosceles Triangle**

**What is the isosceles triangle’s area?**

Half of a triangle’s base plus height is considered to be the area of an isosceles triangle. The area of an isosceles triangle can be calculated using the formula (12) bh square units.

**What is the isosceles triangle’s perimeter?**

Since the isosceles triangle has two equal sides, the perimeter can be calculated using the formula 2a+b units, where “a” stands for the length of the triangle’s two equal legs and “b” for its base.

**Mention the isosceles triangle’s two key characteristics.**

In a triangle, the angles that face the two equal sides are likewise equal.

An isosceles triangle has one base and two equal sides, which are known as the legs and the base, respectively.