Isosceles Triangle

Triangles are a fundamental part of geometry that are classified based on their sides and angles. Among the many triangles, the isosceles triangle is a unique type that has two equal sides and two equal angles. In this guide, we will explore everything about an isosceles triangle, including its definition, properties, formulas, and real-life applications.

1.0What is an Isosceles Triangle?

The basic isosceles triangle meaning is a triangle with two equal legs, as the word is derived from the Greek words Isos (equal) and Skelos (legs). To define isosceles triangle, it is any triangle that has two sides of equal measure. The opposite angles of these equal sides of an isosceles triangle are also of equal measure. This distinct property makes the isosceles triangle essential in geometry. The isosceles triangle property refers to the characteristics that define its shape and angles. The following are some of the properties of isosceles triangles that students should take note of.

• Two equal sides: Two sides of an isosceles triangle will be of equal length.
• Two equal angles: The base angles situated at the opposite of the equal sides are always of the same measure.
• Altitude as a bisector: The altitude drawn from the vertex line to the base dissects the vertex and divides the triangle into two right-angled triangles.
• Congruency: Any two isosceles triangle with the same base and height are congruent. 

2.0Types of Isosceles Triangles

There are three major types of isosceles triangle based on angles. 

• Acute Isosceles Triangle: In this isosceles triangle, both base angles are acute or less than 90 degrees. 
• Right Isosceles Triangle: In this isosceles triangle, one angle is exactly 90 degrees.
• Obtuse Isosceles Triangle: In this isosceles triangle, one angle is obtuse, as it is greater than 90 degrees but less than 180 degrees. 

3.0Isosceles Triangle Theorem

The isosceles triangle theorem states that if two sides are congruent in a triangle, then the angles that are at the opposite of these two sides are also congruent. This theorem is fundamental in proving many geometrical concepts related to triangles.

4.0Perimeter of Isosceles Triangle

The perimeter of isosceles triangle is the sum of all its sides. If an isosceles triangle has two equal lengths, a, and the base length, b, then the formula for the perimeter is:

P = a+a+b

P = 2a+b

5.0Area of an Isosceles Triangle

The area of an isosceles triangle is calculated using the base and the height. The formula for the area of an isosceles triangle is: 

A = (½) x base x height 

Alternatively, using side lengths, we can use Heron's formula:

$A=¼xbx\surd (4a^2–b^2)$

Here, a is the length of the equal sides, and b is the base. 

6.0Isosceles Triangle Angles

Since isosceles triangle angles have two equal angles, knowing one angle helps determine the others. Since we know the sum of the interior angles of a triangle is 180 degrees, it is easy to determine the missing angle in an isosceles triangle. 

If we take an example of isosceles triangle and the vertex angle is 40°, the base angles can be calculated as:

θ = (180° – 40°) / 2 = 70°

7.0Comparison Table of Different Types of Triangles

To understand the differences better, let us compare an isosceles triangle with other types of triangles:

Triangle Type	Side Length	Angle Properties	Example	Images
Equilateral Triangle	All three sides are equal	All three angles are 60°	Triangle with sides 5 cm each
Isosceles Triangle	Two sides are equal	Two equal base angles	Triangle with sides 6 cm, 6 cm, and 4 cm
Scalene Triangle	All sides have different lengths	All angles are different	Triangle with sides 5 cm, 7 cm, and 9 cm
Right Triangle	One right angle	Follows Pythagoras theorem	Triangle with sides 3 cm, 4 cm, and 5 cm

8.0Real-Life Applications of Isosceles Triangles

Isosceles triangles have many real-life applications beyond the boundaries of textbooks. It appears in various fields, including: 

• Architecture: Roof trusses and bridges use isosceles properties for stability.
• Engineering: Isosceles triangles are used to design ramps and other support structures. 
• Art and Design: Isosceles triangles are seen in many patterns and decorations.
• Physics: Light reflection in optics often involves isosceles triangles. 

9.0Solved Examples

Problem 1: Given an isosceles triangle with equal sides of 6 cm and a base of 8 cm.

Find the perimeter and area of this triangle.

Solution: 

Given that the length of the equal sides of an isosceles triangle is 6cm, with a base equal to 8cm. 

Now, to find the perimeter of this triangle: 

P = 6 + 6 + 8 = 20 cm

To find the area:

To use the height formula: 

h = √(a² - (b/2)²) = √(6² - 4²) = √(36 - 16) = √20 ≈ 4.47 cm

A = (1/2) × 8 × 4.47 = 17.88 cm²

Problem 2: Given an isosceles triangle with equal sides of 6 cm and a base of 8 cm.

Solution: 

• To find the perimeter: 

P = 6 + 6 + 8 = 20 cm

• To find the area:

To use the height formula: 

h = √(a² - (b/2)²) = √(6² - 4²) = √(36 - 16) = √20 ≈ 4.47 cm

A = (1/2) × 8 × 4.47 = 17.88 cm²

Problem 3: An isosceles triangle has a base of 10 cm and equal sides of 13 cm. Find its height

Solution: 

The height (h) bisects the base into two equal parts, making each half 5 cm.

Using the Pythagorean theorem in the right triangle formed:

h² + 5² = 13²

h² + 25 = 169

h² = 144

h = 12 cm.

Answer: The height is 12 cm.

Problem 4: An isosceles triangle has a base of 8 cm and equal sides of 6 cm. Find the perimeter.

Solution: 

Perimeter = 2 × Equal Side + Base

= 2 × 6 + 8

= 20 cm.

Answer: The perimeter is 20 cm.

Problem 5: Find the area of an isosceles triangle with a base of 12 cm and a height of 9 cm.

Solution: 

Area = ½ x base x height

Area = ½ x 12 x 9 = 54 cm²

The area of the isosceles triangle is 54 cm².