Question

Geometry Express the area \(A\) of an isosceles right triangle as a function of the length \(x\) of one of the two equal sides.

Short answer

The area is \( A = \frac{1}{2} x^2 \).

Step-by-step solution

Understand the Properties of an Isosceles Right Triangle

An isosceles right triangle has two equal sides and one right angle. This means that the two equal sides form the right angle.

Recall the Formula for the Area of a Triangle

The area of any triangle can be calculated using the formula: \( A = \frac{1}{2} \times \text{base} \times \text{height} \)

Identify the Base and Height in the Isosceles Right Triangle

In this isosceles right triangle, since it has two equal sides forming the right angle, we can use these two equal sides as the base and height of the triangle. Thus, the base and the height are both equal to the length of the sides, denoted as \( x \).

Apply the Triangle Area Formula

Substitute the values identified in Step 3 into the area formula: \[ A = \frac{1}{2} \times x \times x \]

Simplify the Expression

Perform the multiplication to simplify the expression: \[ A = \frac{1}{2} x^2 \]

Key concepts

isosceles right triangle

An isosceles right triangle is a special type of triangle. It has two equal sides and one right angle. The right angle is formed by these two equal sides. This means that each of these sides not only defines the length but also helps form the 90-degree angle.
For practical purposes, imagine a triangle where two legs are the same, and they meet at the right angle. This symmetry makes isosceles right triangles particularly simple to analyze in geometry.

triangle area formula

The area of a triangle is the amount of space inside it. To calculate it, you can use a simple formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this formula, 'base' is the length of one side of the triangle, and 'height' is the perpendicular distance from this side to the opposite vertex.
However, this formula is versatile and applies to any type of triangle. It's particularly easy to use for right triangles because their height is one of their sides.

expressing area as a function

Using the triangle area formula, we can express the area of an isosceles right triangle as a mathematical function of one of its sides.
Since both the base and the height of an isosceles right triangle are equal to the length of its sides, we substitute the side length into the formula.
So, for an isosceles right triangle with side length \( x \), the area \[ A = \frac{1}{2} \times x \times x = \frac{1}{2} x^2 \]
This means that the area of an isosceles right triangle can be represented simply as a function of the side length, showing a direct relationship.

geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and properties of space. In the context of our problem, understanding basic geometric principles helps solve for the area of an isosceles right triangle.
In geometry, triangles are fundamental shapes. Knowing the properties of different types of triangles, like the isosceles right triangle, can simplify complex problems.

• Isosceles triangles have two equal sides.
• Right triangles have a 90-degree angle.
• Combining these properties makes it easier to use familiar formulas and solve for unknown measures.

By learning these principles, you can better understand and navigate the world of geometry and its applications.