Question

Express each statement as an equation involving the indicated variables. Area of a Triangle The area \(A\) of a triangle is one-half the product of its base \(b\) and its height \(h\)

Short answer

A = \frac{1}{2} b h

Step-by-step solution

- Identify Key Variables

Identify the variables involved in the problem. In this case, the variables are the area of the triangle (A), the base of the triangle (b), and the height of the triangle (h).

- Understand the Relationship

Understand the mathematical relationship between the area of a triangle, its base, and its height. The problem states that the area is one-half the product of the base and the height.

- Formulate the Equation

Using the relationship from Step 2, write the equation. The area of the triangle is one-half of the product of its base and height. Therefore, the equation is: \(A = \frac{1}{2} b h\)

Key concepts

triangle area

Triangles are basic geometric shapes but calculating their area might seem tricky at first. However, it's pretty straightforward once you know the formula. The area of a triangle, denoted as \(A\), is essentially the measure of the space enclosed within its three sides. This space can be determined if you know the length of the base and the height of the triangle. Calculating the area of a triangle is a fundamental skill in geometry, applied in many real-world scenarios like construction and design. So, what is the formula? Let's delve into that next.

base and height relationship

To find the area of a triangle, you need to understand the relationship between the base and the height. The base \(b\) can be any one of the triangle's sides, but it's most common to choose the bottom side of a triangle for simplicity. The height of the triangle \(h\) is the perpendicular distance from the chosen base to the opposite vertex (the highest point directly above or below the base). This relationship is crucial because the formula for calculating area involves these two measurements. In simpler terms, picture it like this: the base gives you the length, and the height gives you how tall the triangle stands.

geometric formulas

Now let's incorporate our understanding of the base and height into a geometric formula to calculate the area of a triangle. The formula is quite simple: the area \(A\) of a triangle is equal to one-half the product of its base \(b\) and height \(h\). Mathematically, it looks like this: \[A = \frac{1}{2} b h \]. Let's break this down: \

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• \(b \cdot h\): First, you multiply the base by the height. This gives you twice the area of the triangle, as if there were a mirrored triangle placed against itself.
\
• \( \frac{1}{2} \): Then, you multiply the result by \( \frac{1}{2} \). This accounts for the fact that a triangle is half of a parallelogram formed by doubling its figure.
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By following these steps, you can compute the area of any triangle as long as you know its base and height.