Introduction
When dealing with geometric shapes, especially triangles, understanding their properties and measurements is essential. One such triangle type that often appears in mathematical problems is the isosceles right triangle. This article aims to explore how to calculate the height of an isosceles right triangle given the length of its legs. We will guide you through a step-by-step solution using the Pythagorean Theorem, ensuring clarity and accuracy in the process.
Understanding the Problem
Given an isosceles right triangle, where the two legs are of equal length (20 cm each), we need to find the height (or altitude) of the triangle. The problem often becomes complicated when insufficient information is provided, such as the base or the hypotenuse length. In this context, we will focus on the correct application of the Pythagorean Theorem and provide a detailed solution.
Solution
Part I: Finding the Hypotenuse
First, we need to find the hypotenuse of the isosceles right triangle. The Pythagorean Theorem states that for a right triangle with legs a and b, and hypotenuse c, the following equation holds:
c^2 a^2 b^2
Given that the legs a and b are both 20 cm, we can substitute these values into the equation:
c^2 20^2 20^2 400 400 800
Therefore, the hypotenuse c is:
c sqrt{800} sqrt{400 times 2} 20sqrt{2} , text{cm}
Part II: Finding the Height
The height (or altitude) of an isosceles right triangle can be found by dividing the hypotenuse into two equal segments, each being half of the hypotenuse. Let us denote the half-hypotenuse as H':
H' frac{c}{2} frac{20sqrt{2}}{2} 10sqrt{2} , text{cm}
Now, consider the right triangle formed by the height, half of the hypotenuse, and the leg of the isosceles triangle. We can apply the Pythagorean Theorem again to find the height A:
20^2 A^2 (10sqrt{2})^2
Simplifying further, we get:
400 A^2 100 times 2
400 A^2 200
A^2 200
A sqrt{200} sqrt{100 times 2} 10sqrt{2} , text{cm}
Hence, the height of the isosceles right triangle is 10sqrt{2} cm, approximately 14.14 cm.
Conclusion
In conclusion, the height of an isosceles right triangle with legs of 20 cm each is 10sqrt{2} cm, or approximately 14.14 cm. This solution demonstrates the application of the Pythagorean Theorem to solve geometric problems accurately. It is important to ensure that all necessary measurements and conditions are provided to achieve a precise answer.