Problem 80 - Algebra for College Students

Question

Set up an equation and solve each of the following problems. The length of an altitude of a triangle is one-third the length of the side to which it is drawn. If the area of the triangle is 6 square centimeters, find the length of that altitude.

Step-by-Step Solution

Answer

The length of the altitude is 2 cm.

Step 1: Define Variables

Let the length of the altitude be \( h \) and the side to which it is drawn be \( s \). According to the problem, \( h = \frac{1}{3}s \).

Step 2: Write Formula for Triangle Area

The area \( A \) of a triangle is given by the formula: \( A = \frac{1}{2} \times \, \text{base} \times \text{height} \). Here, the base is \( s \) and the height is \( h \). The area is given as 6 cm².

Step 3: Substitute Values into Area Formula

Substitute the given area and the expression for \( h \): \[ 6 = \frac{1}{2} \times s \times \frac{1}{3}s \]

Step 4: Simplify the Equation

Simplify the expression: \[ 6 = \frac{1}{6}s^2 \], which implies \[ s^2 = 36 \].

Step 5: Solve for Side Length \( s \)

Solve for \( s \) by taking the square root of both sides: \[ s = \sqrt{36} = 6 \].

Step 6: Find the Altitude Length

Recall \( h = \frac{1}{3}s \). Substitute \( s = 6 \): \[ h = \frac{1}{3} \times 6 = 2 \].

Key Concepts

Triangle Area

The area of a triangle is one of the basic concepts of geometry that helps in understanding the size of a triangle's surface. To find the area, you will need to know the base and the height (also known as altitude) of the triangle. The formula is simple yet powerful:

• The area of a triangle is calculated as: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

This means you take half of the product of the base length and the height. In our given problem, the base is represented by \( s \) and the height is \( h \).
Understanding this formula is essential because it not only applies to simple problems but also plays a crucial role in more complicated geometric and algebraic situations. In our exercise, the known area helped us set up an equation to solve for unknown dimensions of the triangle.
Remember, the base and height must be perpendicular to each other. Without this relationship, the area cannot be accurately calculated.

Altitude of a Triangle

The altitude of a triangle, also known as the height, is an important concept in geometry. It refers to the perpendicular segment from a vertex to the line containing the opposite side.
In solving problems related to altitude, setting up a relationship between different parts of the triangle can help simplify calculations.

• For example, in this exercise, the altitude is said to be one-third the length of the base. This translates to the equation: \( h = \frac{1}{3}s \).

This kind of relationship can be used to find unknown measurements when combined with other information, such as the area.
Once we calculated \( s \) as 6 cm using the area formula, we easily determined \( h \) as 2 cm.
Altitudes are not just crucial for determining area; they are also useful for other topics like triangle congruence and similarity.

Geometric Problem Solving

Geometric problem solving involves using equations and formulas to find unknown values. In our exercise, we used the relationship between the altitude, the base, and the area of a triangle to set up an algebraic equation.

• We started by translating the given relationship \( h = \frac{1}{3}s \) into mathematical terms.
• Next, we applied the area formula \( A = \frac{1}{2} \times s \times h = 6 \) to substitute and form an equation: \( 6 = \frac{1}{2} \times s \times \frac{1}{3}s \).
• By simplifying, we reduced this to a simpler equation: \( s^2 = 36 \), from which we found \( s \).

Breaking down a complex problem into smaller, manageable parts is key to successful problem-solving.
With practice, you'll become adept at identifying relationships and strategically using different formulas to find solutions.