Perimeter Of Isosceles Right Angled Triangle

Understanding the Perimeter of an Isosceles Right-Angled Triangle

Finding the perimeter of any triangle is straightforward: simply add the lengths of all three sides. However, the isosceles right-angled triangle presents a unique opportunity to explore a more concise formula due to its specific properties. This article will guide you through understanding these properties and calculating the perimeter efficiently. We'll cover various approaches and provide examples to solidify your understanding.

What is an Isosceles Right-Angled Triangle?

An isosceles right-angled triangle is a special type of triangle possessing two key characteristics:

• Right-angled: It contains one angle that measures exactly 90 degrees.
• Isosceles: Two of its sides are equal in length. These two equal sides are the legs, while the side opposite the right angle (the longest side) is the hypotenuse.

This combination of properties leads to predictable relationships between the sides, making perimeter calculations simpler.

Calculating the Perimeter: Different Approaches

There are several ways to determine the perimeter, depending on the information given:

1. Knowing the Length of the Legs:

This is the most straightforward method. Since two sides (legs) are equal, let's call their length 'a'. The hypotenuse ('c') can be calculated using the Pythagorean theorem: c² = a² + a² = 2a². Therefore, c = a√2.

The perimeter (P) is then: P = a + a + a√2 = 2a + a√2 = a(2 + √2)

Example: If the legs of an isosceles right-angled triangle are each 5 cm long, then:

• a = 5 cm
• c = 5√2 cm
• P = 5(2 + √2) ≈ 19.07 cm

2. Knowing the Length of the Hypotenuse:

If you know the hypotenuse ('c'), you can work backward to find the length of the legs. Since c = a√2, then a = c/√2. Rationalizing the denominator, we get a = c√2/2.

The perimeter (P) is then: P = (c√2/2) + (c√2/2) + c = c√2 + c = c(1 + √2)

Example: If the hypotenuse of an isosceles right-angled triangle is 10 cm long, then:

• c = 10 cm
• a = 10√2/2 = 5√2 cm
• P = 10(1 + √2) ≈ 24.14 cm

3. Knowing the Area:

While less direct, the area (A) can also be used. The area of a right-angled triangle is (1/2) * base * height. In an isosceles right-angled triangle, the base and height are equal to the leg length 'a'. So, A = (1/2)a².

Solving for 'a', we get a = √(2A). Then, you can use the formula from method 1 to calculate the perimeter.

In summary: The most efficient approach depends on the information provided. Remember that understanding the relationship between the legs and the hypotenuse (a√2) is crucial for solving problems involving isosceles right-angled triangles. This simple relationship significantly streamlines the perimeter calculation.