Problem 15 - College Algebra with Modeling and Visualization

Question

Find the area of the triangle with base \(b\) and height \(h .\)

Step-by-Step Solution

Answer

The area of the triangle is given by \( A = \frac{b \times h}{2} \).

Step 1: Understanding the Formula for Area of a Triangle

The area of a triangle can be found using the formula \( A = \frac{1}{2} \times b \times h \), where \( b \) is the base of the triangle and \( h \) is the height.

Step 2: Substituting Known Values

We substitute the given values of the base \( b \) and height \( h \) into the formula. Thus, the area \( A \) becomes \( A = \frac{1}{2} \times b \times h \).

Step 3: Calculating the Area

Proceed to compute the area by performing the multiplication and division specified in the formula. This results in \( A = \frac{b \times h}{2} \).

Key Concepts

Base and Height of a Triangle

When calculating the area of a triangle, it’s crucial to identify the base and the height. The base (\( b \)) of a triangle is any one of its three sides, usually the one that lies horizontally at the bottom.
The height (\( h \)) is a vertical line that extends from the base to the opposite vertex, forming a right angle with the base. This perpendicularity is key because the height must always be the shortest distance to the base.

When identifying these values in a triangle:

• Base can be any side of the triangle. However, it's often more practical to use the ground-facing side as the base.
• Height will always be measured in a straight line from the base to the top angle, ensuring right angles are formed.

These two measurements, base and height, lay the foundation for calculating the triangle’s area using the well-known formula.

Triangle Area Formula

The formula to calculate the area of a triangle is both straightforward and powerful. It is expressed as:\[ A = \frac{1}{2} \times b \times h \]where:

• \( A \) is the area of the triangle
• \( b \) is the length of the base
• \( h \) is the height of the triangle

This formula essentially breaks down to multiplying the base and the height and then dividing the result by two. The division by two is because a triangle is essentially half of a rectangle when both share the same base and height.

This straightforward calculation allows you to determine the triangular area simply and accurately once you have your base and height measurements.

Mathematical Problem Solving

Solving problems involving triangle areas is an exercise in applying mathematical concepts systematically. Here’s a structured approach:

• Identify Known Variables: Always start by determining the base and the height of the triangle from the problem statement.
• Apply the Formula: Substitute the identified values into the area formula \( A = \frac{1}{2} \times b \times h \).
• Solve: Calculate the multiplication of base and height, then divide the resulting product by 2 to get the area.

Breaking the problem into these steps simplifies complex calculations. With practice, it becomes intuitive to follow these methods, making triangle area problems more manageable for students.