What is the area of a 45-45-90 triangle?
Understand the Problem
The question is asking for the area of a 45-45-90 triangle, which is a special type of isosceles right triangle where the angles are 45 degrees. To find the area, we can use the formula for the area of a triangle: Area = (1/2) * base * height. In a 45-45-90 triangle, the base and height are equal, so we can express the area in terms of one of the equal sides.
Answer
The area of a 45-45-90 triangle is given by the formula: $$ \text{Area} = \frac{1}{2} x^2 $$
Answer for screen readers
The area of a 45-45-90 triangle with legs of length $x$ is given by the formula: $$ \text{Area} = \frac{1}{2} x^2 $$
Steps to Solve
- Identify the sides of the triangle
In a 45-45-90 triangle, the two legs (base and height) are of equal length. Let's denote the length of each leg as $x$.
- Apply the area formula for a triangle
The formula for the area of a triangle is given by: $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$
Since the base and height are both equal to $x$, we can substitute: $$ \text{Area} = \frac{1}{2} \times x \times x $$
- Simplify the area formula
Now, simplify the equation: $$ \text{Area} = \frac{1}{2} x^2 $$
- Substitute a specific value for x (if given)
If a specific length for the legs is provided, substitute that value in place of $x$ to calculate the area. For instance, if each leg measures 5 units, then: $$ \text{Area} = \frac{1}{2} \times 5^2 = \frac{1}{2} \times 25 = 12.5 $$
The area of a 45-45-90 triangle with legs of length $x$ is given by the formula: $$ \text{Area} = \frac{1}{2} x^2 $$
More Information
A 45-45-90 triangle is a special type of isosceles right triangle where the angles are equal. The relationship between the sides allows for easy calculation of the area. If you know the length of either leg, you can quickly determine the area using the formula derived above.
Tips
- Forgetting that the base and height are equal in a 45-45-90 triangle.
- Confusing the formula for area with other triangle types; always use the correct formula for the triangle you are working with.