Calculating the Sides of an Isosceles Right-Angled Triangle with a Given Hypotenuse
In this article, we will explore how to find the lengths of the sides of an isosceles right-angled triangle when the hypotenuse is given. Specifically, we will calculate the lengths of the legs of a triangle with a hypotenuse of 50 units.
Introduction to Isosceles Right-Angled Triangles
An isosceles right-angled triangle is a special type of triangle where two sides are equal and the angle between them is 90 degrees. In such triangles, the lengths of the sides can be derived from the Pythagorean theorem and the properties of 45-degree angles.
Using the Cosine and Sine Functions
Given that the two equal angles are each 45 degrees, we can use trigonometric functions:
Hypotenuse (c) 50
Other two sides (a and b) c * cos(45) c * sin(45) 50 * cos(45) 50 * sin(45)
Since cos(45) and sin(45) are both equal to √2 / 2:
Side length 50 * (√2 / 2) 50 / √2 ≈ 35.35533906 cm
The Pythagorean Theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
c^2 a^2 b^2
Given the hypotenuse c 50, we can calculate the sides:
50^2 2 * a^2 (since a b in an isosceles triangle)
2500 2a^2
a^2 1250
a √1250 ≈ 35.3553390593274 cm
Proportional Method
In an isosceles right-angled triangle, the sides are in the ratio 1:1:√2. Given that the hypotenuse is 50:
Let the sides be denoted as a. Then:
2a^2 50^2
2a^2 2500
a^2 1250
a 50 / √2 25√2
Conclusion
The lengths of the sides of an isosceles right-angled triangle with a hypotenuse of 50 units can be calculated using trigonometric functions, the Pythagorean theorem, or a proportional method. In all cases, the lengths of the legs are approximately 35.35533906 cm.
Additional Resources
For more in-depth understanding of triangle properties and calculations, consider exploring the following resources:
Math is Fun - Triangle Types Khan Academy - Pythagorean Theorem - Isosceles TrianglesFAQ
What is the relationship between the sides of an isosceles right-angled triangle?
The sides are in the ratio 1:1:√2, where the hypotenuse is √2 times the length of one of the legs.
How do you find the sides of a right-angled triangle?
Use the Pythagorean theorem (a^2 b^2 c^2), trigonometric functions, or the known proportion to find the sides.
Why are 45-degree angles important in right-angled triangles?
45-degree angles make an isosceles right-angled triangle, where the two legs are equal, and the hypotenuse can be easily calculated using trigonometric functions or the Pythagorean theorem.