Problem 34 - Algebra and Trigonometry

Question

Write an algebraic formula for the given quantity. You may need to consult the formulas for area and volume listed on the inside front cover of this book. The area \(A\) of a triangle whose base is twice its height \(h\)

Step-by-Step Solution

Answer

The area of the triangle is \( h^2 \).

Step 1: Understand the Problem

We are given a triangle where the base is twice its height. We need to express the area of this triangle in terms of its height, using the formula for the area of a triangle.

Step 2: Identify the Area Formula

The formula for the area of a triangle is \[ A = \frac{1}{2} \times \text{base} \times \text{height}. \] In this case, we will use this formula to express the area given the relationship between the base and height.

Step 3: Express the Base in Terms of Height

According to the problem, the base of the triangle is twice its height. Hence, \[ \text{base} = 2h. \]

Step 4: Substitute the Base into the Area Formula

Substitute \( 2h \) for the base in the area formula:\[ A = \frac{1}{2} \times 2h \times h. \]

Step 5: Simplify the Expression

Simplify the expression for the area:\[ A = \frac{1}{2} \times 2h^2 = h^2. \] Thus, the area of the triangle in terms of height is \( h^2 \).

Key Concepts

Triangle Area

When calculating the area of a triangle, it is essential to understand the role of its base and height. The area represents the space enclosed by the three sides of the triangle. More technically, the area can be determined using the formula:
- \[A = \frac{1}{2} \times \text{base} \times \text{height}\]This formula helps us find the area when we know the base and height. A useful way to remember this is that you're essentially calculating half of the parallelogram that would be formed if you had two identical triangles put together.
In the original exercise, the base is twice as long as the height. By identifying this relationship, we can express areas more directly using algebra.

Geometric Formulas

Geometric formulas are mathematical formulas that are used to calculate measurements and relationships between different dimensions of geometric shapes. They are foundational in both pure and applied mathematics.
Some important geometric formulas account for:

• Area
• Perimeter
• Volume

In our exercise, the student refers to a classic geometric formula for the area of a triangle. Understanding these formulas enables students to move from theoretical math knowledge to practical application. Recognizing the relationship in triangles where one dimension depends on another is a stepping stone in advanced mathematics.
When given such relationships, one should always start by clearly identifying each quantity involved and how it relates to others. Using the triangle area formula connects our understanding of geometry to algebra, demonstrating how these concepts intersect.

Expression Simplification

Expression simplification refers to the process of making a mathematical expression easier to read and understand without changing its value. It may involve combining like terms, reducing fractions, or applying algebraic identities.
For our specific problem, after substituting the base's expression into the area formula, simplification becomes essential. This step helps reduce complex expressions into understandable formats. In this case, our initially long expression:

• \[\frac{1}{2} \times 2h \times h\]

Can be simplified to:

• \[h^2\]

This is achieved by first multiplying the terms,
then recognizing the cancellation of common factors such as 2 in the numerator and denominator. Mastering simplification aids in solving algebraic problems more efficiently, is crucial in upper-level math, and improves computational fluency. It's a vital skill that allows mathematicians to communicate their results clearly.