Problem 9 - Introductory Algebra for College Students

Question

A triangle has a base of 5 feet and an area of 20 square feet. Find the triangle's height.

Step-by-Step Solution

Answer

The height of the triangle is 8 feet.

Step 1: Understand the Problem

The problem provides the base and area of a triangle and requires finding the height. This problem is a straightforward application of the formula for the area of a triangle.

Step 2: Write down the known values

From the problem, the area \( A \) of the triangle is known to be 20 square feet and the base \( b \) is known to be 5 feet.

Step 3: Use the Triangle Area formula to find the height

The formula for the area of a triangle is generally given as \( A = \frac{1}{2}*b*h \), where \( A \) is the area, \( b \) is the length of the base, and \( h \) is the height. Rearrange this formula to solve for the height \( h \): \( h = \frac{2*A}{b} \). Substituting the known values gives \( h = \frac{2*20}{5} \).

Step 4: Calculate the height

Carry out the division to find the height: \( h = \frac{40}{5} = 8 \) feet

Key Concepts

Triangle Area Formula

Triangles are fascinating shapes in geometry, often used in various mathematical problems to determine unknown measurements. The area of a triangle can be calculated using a specific formula, which finds usefulness in different scenarios. The formula for the area of a triangle is given as:

• \( A = \frac{1}{2} \times b \times h \)

where:

• \( A \) represents the area of the triangle
• \( b \) stands for the base of the triangle
• \( h \) is the height of the triangle

This formula is crucial as it allows us to find either the area, base, or height of a triangle if at least two other measurements are known. It underscores the relationship between the base, height, and resulting area. This foundational concept is key for students, providing a simple yet powerful tool for solving real-world problems involving triangles.

Solving for Height

One of the common uses of the triangle area formula is to solve for the height of a triangle when the area and the base are known. To do this, we rearrange the standard area formula \( A = \frac{1}{2} \times b \times h \) to solve for height \( h \). The equation becomes:

• \( h = \frac{2A}{b} \)

This operation involves basic algebraic manipulation, which is rearranging equations to isolate the variable of interest. We multiply both sides by 2 to eliminate the fraction, and then divide by the base \( b \) to solve for \( h \), the height. Returning to the original problem's context, we know the area \( A = 20 \) square feet and the base \( b = 5 \) feet, so substituting these values into our rearranged equation gives:

• \( h = \frac{2 \times 20}{5} = 8 \) feet

This demonstrates how rearranging formulas and applying known values allows us to solve practical geometry problems efficiently.

Basic Geometry Concepts

Understanding basic geometry concepts is essential for solving many problems in mathematics, such as the one given in this exercise. Let's break down some core ideas:

• **Geometric Shapes:** Triangles are simple polygons with three sides and are classified based on their angles and sides. Different types of triangles include scalene, isosceles, and equilateral triangles.
• **Measurements:** In geometry, clear definitions of measurements like area, perimeter, and volume are required for solving problems involving shapes. The area measures the space inside a shape, perimeter measures the boundary length, and volume measures space within a 3D object.
• **Formulas:** Formulas are key in geometry because they provide quick methods to calculate dimensions and properties of shapes. Knowing formulas for shapes like triangles can save time and simplify complex problem-solving processes.

Grasping these basic concepts allows students to build a robust foundation in geometry, making it easier to tackle both simple and complex mathematical challenges effectively.