Problem 45 - College Algebra

Question

If the area model for a triangle is \(A=\frac{1}{2} b h,\) find the area of a triangle with a height of 16 in. and a base of 11 in.

Step-by-Step Solution

Answer

The area of the triangle is 88 square inches.

Step 1: Identify the Formula

The problem provides the formula to find the area of a triangle as \(A = \frac{1}{2} b h\), where \(b\) is the base length and \(h\) is the height of the triangle.

Step 2: Substitute Given Values

The triangle's height is 16 inches and the base is 11 inches. Substitute these values into the formula: \(A = \frac{1}{2} \times 11 \times 16\).

Step 3: Compute the Product

First, calculate the product of the base and height: \(11 \times 16 = 176\).

Step 4: Calculate the Area

Multiply the result by \(\frac{1}{2}\): \(A = \frac{1}{2} \times 176 = 88\).

Step 5: Provide the Final Answer

The area of the triangle is thus \(88\) square inches.

Key Concepts

Understanding Geometry and Its Importance

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space.
It helps us understand the physical world around us and is essential for various fields such as engineering, architecture, and art.
This branch of mathematics involves studying points, lines, angles, surfaces, and solids.

In our everyday lives, knowing geometry can help us solve practical problems.

• For instance, when you're building something or arranging furniture, you apply geometric concepts.
• It's also crucial for design-related professions, where space management is key.
• Geometry also plays a significant role in sports; for instance, calculating the best angles for shooting or passing.

Recognizing shapes and understanding how they interact within a space enhances spatial awareness and critical thinking skills.

The Triangle Area Formula Explained

The area of a triangle is a measure of the space within the triangle's boundaries.
For calculating this area, we use the triangle area formula:
Given by: \[ A = \frac{1}{2} bh \]
Where:

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

This formula is derived from the area of a rectangle, as a triangle with the same base and height has half the area of the enclosing rectangle.
Having the correct measurement units is crucial, typically with the area in square units such as square inches, square centimeters, etc.
This understanding lays the groundwork for more complex geometric concepts later on.

Steps for Calculating the Area of a Triangle

To calculate the area of a triangle, follow these simple steps:

First, identify the base and height of the triangle. These dimensions must be perpendicular to each other.
Once you have these measurements, substitute them into the area formula:
\( A = \frac{1}{2} b h \)
Then, calculate the product of the base and height.
This step consolidates the triangle's dimensions into a unified metric.
Finally, multiply that value by \( \frac{1}{2} \) to find the area.
Essentially, this step divides the rectangle's area, which would surround the triangle, by half.
In the example provided, with a height of 16 inches and a base of 11 inches:
- Multiplying the base and height gives 176.
- Taking half of this product gives the final area of 88 square inches.
Understanding these steps can make geometry less intimidating and more approachable for any student.