Problem 219 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. A triangular flag has base one foot and height 1.5 foot. What is its area?

Step-by-Step Solution

Answer

0.75 square feet

Step 1: Identify the Formula for Area of a Triangle

The area of a triangle is given by the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

Step 2: Substitute the Given Values

We know the base is 1 foot and the height is 1.5 foot. Substitute these values into the formula: \[ \text{Area} = \frac{1}{2} \times 1 \times 1.5 \]

Step 3: Calculate the Area

Perform the multiplication and division: \[ \text{Area} = \frac{1}{2} \times 1.5 = 0.75 \text{ square feet} \]

Key Concepts

Triangle Properties

When studying the area of a triangle, it's essential to understand the fundamental properties of triangles. Triangles come in different types such as equilateral, isosceles, and scalene, each with unique characteristics.

• Equilateral triangle: All three sides and angles are equal.
• Isosceles triangle: Has two sides of equal length and two equal angles.
• Scalene triangle: All three sides and angles are different.

Each type of triangle has the same basic property: the sum of the internal angles is always 180 degrees. Knowing these properties helps you understand more advanced concepts related to triangles.

Geometry

Geometry is the branch of mathematics that explores the properties and relations of points, lines, surfaces, and shapes, including triangles. In geometry, triangles are one of the basic shapes studied. The main components of a triangle include the base, height (or altitude), and the sides.
The base of a triangle can be any one of its sides, but is typically chosen based on where the height is drawn. The height is the perpendicular distance from the chosen base to the opposite vertex.
Understanding these geometrical elements is crucial before diving into calculations like finding the area.

Formula Substitution

To find the area of a triangle, we use the formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
It's essential first to identify your base and height in the given problem.

• In this exercise, the base is provided as 1 foot.
• The height is given as 1.5 feet.

Substituting these values into the formula helps you simplify the problem step-by-step. Properly substituting values ensures accurate calculations and prevents mistakes.

Area Calculation

Finally, it's time to calculate the area after substituting values. Using our formula: \( \text{Area} = \frac{1}{2} \times 1 \times 1.5 \).
Simplify inside the parentheses first, starting with the multiplication: \( 1 \times 1.5 = 1.5 \).
Then multiply by \( \frac{1}{2} \): \( \frac{1}{2} \times 1.5 = 0.75 \).
Therefore, the area of the triangular flag is 0.75 square feet. Breaking down the operations step-by-step ensures you make no errors and understand each part of the calculation process.