Problem 86 - Algebra 1: Concepts and Skills

Question

Find the area of a triangle with the given base and height. $$b=4, h=9$$

Step-by-Step Solution

Answer

The area of the triangle is 18 square units.

Step 1: Identify the given values

In this exercise, the base \(b\) of the triangle is given as 4 and the height \(h\) as 9.

Step 2: Apply the formula for the area of a triangle

The formula to compute the area of a triangle is \( Area = \frac{1}{2} \times base \times height \) . Substituting the given values, it becomes \( Area = \frac{1}{2} \times 4 \times 9 \)

Step 3: Compute the Area

Perform the multiplication and division operations to get the Area. \(Area = \frac{1}{2} \times 4 \times 9 = 2 \times 9 = 18 \) square units

Key Concepts

Geometry: Understanding Shapes and Spaces

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. It is vital for understanding how to calculate various measurements, such as the area of different shapes. The concept of area is fundamental in geometry, as it represents the size of a two-dimensional surface. For instance, the exercise requires the calculation of the area of a triangle, which is a basic geometric shape.

A triangle has three sides and is known for its three angles, adding up to 180 degrees. The ability to calculate a triangle's area is not just an academic exercise; it permeates into various real-world applications, such as architecture and engineering.

To calculate a triangle's area, the essential measurements are its base and height. The base is one side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex. Remember, the terms 'base' and 'height' can apply to any of the three sides and their respective perpendiculars, not just what may initially appear as the 'bottom' or 'sideways' triangle.

Mathematical Formulas: The Language of Calculation

Mathematical formulas are like a shorthand language that provides an efficient way to represent real-world problems and their solutions. Formulas are crucial because they give a consistent and systematic method to solve similar problems. In this exercise, the formula for finding the area of a triangle \( A = \frac{1}{2} \times b \times h \) is used, where \( b \) is the base, and \( h \) is the height.

This specific formula is an example of how mathematical expressions encapsulate a rule for problem-solving. It simplifies the process, ensuring you don't have to understand the underlying proof every time you calculate an area—though understanding the proof is beneficial for deeper learning.

It is also a perfect example of how we can use algebra to generalize a solution, since the formula is applicable for any values of base and height. This universality is powerful; it means that you can apply the same steps to a vast array of problems, which is an efficient way to learn and apply mathematical concepts.

Problem-Solving in Algebra: From Known to Unknown

Problem-solving in algebra involves finding unknown values using the given information and relationships between variables. The step-by-step solution to finding the area of a triangle is an application of problem-solving in algebra. First, you identify the known values (the base and height), and then you use the algebraic formula to find the unknown value (the area).

Prioritizing information is key when approaching algebraic problems. Identifying what you know and what you need to find out can guide your use of formulas. In our problem, the known values, \( b = 4 \) and \( h = 9 \) are substituted into the area formula for a triangle. In doing so, you're transforming the abstract formula into a concrete solution.

Finally, solving the algebraic expression \( A = \frac{1}{2} \times 4 \times 9 \) involves operations of multiplication and division, which are fundamental in algebra and lead you to the final answer, 18 square units. This showcases how algebra serves as the underlying toolset for turning formulas and known values into solutions.