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45-45-90 Triangle: Finding Missing Sides
In geometry, a 45-45-90 triangle is a special type of right triangle where two of its angles measure 45 degrees each. This unique configuration leads to a specific relationship between the sides of the triangle, making it easier to calculate missing side lengths.
Understanding the Special Ratio
The key to solving 45-45-90 triangles lies in the special ratio of its sides. If one leg of the triangle has a length of 'x', the other leg will also have a length of 'x', and the hypotenuse (the side opposite the right angle) will have a length of 'x√2'.
Here's a visual representation:
Using the Pythagorean Theorem
The Pythagorean Theorem, a fundamental principle in geometry, states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem can be applied to 45-45-90 triangles to find missing sides.
Let's say we know the length of one leg is 'a'. Using the Pythagorean Theorem, we can find the length of the hypotenuse (c):
a² + a² = c²
2a² = c²
c = √(2a²)
c = a√2
Example Problem
Let's say we have a 45-45-90 triangle where one leg measures 5 units. We can use the special ratio and the Pythagorean Theorem to find the other leg and the hypotenuse:
Step 1: Find the other leg
Since it's a 45-45-90 triangle, the other leg will also have a length of 5 units.
Step 2: Find the hypotenuse
Using the special ratio, the hypotenuse will be the length of a leg multiplied by √2:
Hypotenuse = 5√2 units
Practice Problems
Try these practice problems to solidify your understanding:
- If one leg of a 45-45-90 triangle is 8 units long, what is the length of the other leg and the hypotenuse?
- If the hypotenuse of a 45-45-90 triangle is 10√2 units long, what is the length of each leg?
Conclusion
Understanding the special ratio and the Pythagorean Theorem makes solving 45-45-90 triangles a breeze. Remember, these triangles have two equal angles and a specific relationship between their sides. Practice using these methods to become proficient in working with this unique type of right triangle.