Problem 222 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. What is the height of a triangle with area 893 square inches and base 38 inches?

Step-by-Step Solution

Answer

The height of the triangle is 47 inches.

Step 1: Recall the Area Formula

The area of a triangle can be found using the formula area = \( \frac{1}{2} \times \text{base} \times \text{height} \).

Step 2: Substitute the Given Values

Substitute the given area and base into the formula: 893 = \( \frac{1}{2} \times 38 \times \text{height} \).

Step 3: Solve for Height

To find the height, solve for height (h): 1. Multiply both sides by 2 to clear the fraction: 1786 = 38 \times \text{height}. 2. Divide both sides by 38: \( \text{height} = \frac{1786}{38} \).

Step 4: Calculate the Height

Divide 1786 by 38: \( \text{height} = 47 \) inches.

Key Concepts

area of a triangle

The area of a triangle is a measure of the space enclosed by the three sides of the triangle. It is a fundamental concept in geometry. The formula to calculate the area of a triangle is \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). This formula is very important and can be easily remembered by noting that it involves multiplying the base and the height, then taking half of that product. The base of a triangle is any one of its sides, typically the one that lies horizontally.

The height (or altitude) of a triangle is a perpendicular line from the base to the opposite vertex. This formula works for all types of triangles, regardless of their shape. Understanding how to use this formula can help you solve many problems involving triangle measurements.

geometry

Geometry is the branch of mathematics dealing with shapes, sizes, and the properties of space. It helps us understand spatial relationships and properties of objects in various dimensions. Triangles are some of the simplest geometric shapes.

Triangles must follow rules related to their angles and sides:

• The sum of the interior angles of a triangle is always 180 degrees.
• The length of any side of a triangle must be less than the sum of the lengths of the other two sides.
• The Pythagorean theorem applies to right triangles, relating the lengths of the legs to the hypotenuse: \(a^2 + b^2 = c^2\).

These properties are fundamental to solving problems in geometry. Recognizing and applying geometric properties allows us to find unknown measurements, just as we've done in our problem with the height of a triangle.

algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It's particularly useful in solving equations where we need to find the value of unknown variables. In our problem, we used algebra to solve for the height of a triangle.

Here's how:

• We first set up the equation based on the area formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
• We substituted the known values (area = 893, base = 38) into the equation to form \( 893 = \frac{1}{2} \times 38 \times \text{height} \).

Next, we cleared the fraction by multiplying both sides by 2: \( 1786 = 38 \times \text{height} \). With this equation, we solved for height by dividing both sides by 38: \( \text{height} = \frac{1786}{38} \). Finally, we calculated the height: \( \text{height} = 47 \text{ inches} \). This step-by-step process shows how algebra helps us find unknown values using known information.