Introduction

Understanding the properties of a geometric shape, such as the isosceles right triangle, can be crucial in various fields, from architectural design to mathematical problem-solving. This article delves into how to find the perimeter of an isosceles right triangle given its hypotenuse. We will explore the mathematical principles involved and provide a step-by-step solution using the Pythagorean theorem.

Isosceles Right Triangle: Basic Properties

An isosceles right triangle is a special type of right triangle where two of the sides are of equal length, and the angles opposite these sides are each 45 degrees. The relationship between the legs (denoted as a) and the hypotenuse (denoted as c) of such a triangle is given by the Pythagorean theorem. Specifically, for an isosceles right triangle, the hypotenuse can be calculated using the formula:

Pythagorean Theorem for Isosceles Right Triangle

c a√2

Calculating Perimeter Given Hypotenuse

Let's consider the case where the hypotenuse c of an isosceles right triangle is 18 dm. To find the perimeter, we first need to determine the length of each leg (a) using the Pythagorean theorem.

Step-by-Step Solution

We know that for an isosceles right triangle, the relationship between the hypotenuse and the legs is given by the formula:c a√2Given that the hypotenuse c is 18 dm, we can express this as:18 a√2To solve for a, divide both sides by √2:a 18/√2 18 × (√2/2) 9√2 dmNow that we have the length of each leg, we can find the perimeter of the triangle. The perimeter (P) is the sum of the lengths of the three sides:P 2a c 2(9√2) 18 18√2 18Approximating √2 as 1.414, we get:P ≈ 18(1.414) 18 ≈ 25.452 18 ≈ 43.452 dm

Generalization

The principles discussed here can be generalized to any isosceles right triangle with a hypotenuse of c dm. The length of each leg would be c/√2, and the perimeter would be calculated as:

General Perimeter Formula

P c√2 c c(√2 1)

Conclusion

In conclusion, understanding and applying the Pythagorean theorem to an isosceles right triangle is a valuable skill. This article has provided a step-by-step guide to finding the perimeter of such a triangle given its hypotenuse. Whether you are a student, a professional, or dealing with geometric problems in your daily life, the knowledge of these principles can be incredibly useful.