Problem 31 - Algebra A Combined Function

Question

The length of the base of a triangle is twice its height. If the area of the triangle is 100 square kilometers, find the height.

Step-by-Step Solution

Answer

The height of the triangle is 10 kilometers.

Step 1: Understand the Problem

The problem gives us the area of a triangle and the relationship between its base and height, and asks us to find the height.

Step 2: Recall the Formula for Area of a Triangle

The area of a triangle can be calculated as \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).

Step 3: Set Up the Equation

Let the height of the triangle be \( h \). According to the problem, the base \( b \) is twice the height, so \( b = 2h \). Substitute these into the area formula: \( 100 = \frac{1}{2} \times 2h \times h \).

Step 4: Simplify the Equation

The equation becomes \( 100 = h^2 \). Divide both sides by 1 to simplify, retaining \( h^2 = 100 \).

Step 5: Solve for the Height

Take the square root of both sides to find \( h \). \( h = \sqrt{100} \).

Step 6: Calculate the Square Root

The square root of 100 is 10, so \( h = 10 \).

Key Concepts

Area of a Triangle

Calculating the area of a triangle is a foundational concept in geometry, and it is pivotal to understand it clearly. The formula used to compute the area of a triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]This formula emerges from the fact that a triangle is essentially half of a parallelogram. Hence, using the base and height, which represent the dimensions of the triangle, allows us to find how much space the triangle takes up in a two-dimensional plane.

• Base: The base refers to the bottom side of the triangle and can be any of the three sides, but is conventionally the side laying horizontally when the triangle is visualized.
• Height: The height is a line segment perpendicular to the base, extending to the opposite vertex of the triangle. Even if the base is changed, the height's measurement will also automatically adapt accordingly, keeping the area calculation correct.

Understanding this interaction between base and height is crucial for efficient and accurate area calculation.

Base and Height Relationship

In some geometric problems, the base and the height of a triangle are related in specific ways, like in our exercise where the base is twice the height. This relationship can simplify problem-solving.

• If the base \( b \) of a triangle is described in terms of the height \( h \), such as \( b = 2h \), you can directly substitute this into the area formula.
• Substituting \( b = 2h \) into the area formula \( \frac{1}{2} \times b \times h \) converts it to \( \frac{1}{2} \times 2h \times h \), simplifying to \( h^2 \).

This relationship helps reduce variables and simplifies the problem to a single equation involving one unknown. Recognizing these relationships will improve your ability to tackle various geometric problems efficiently.

Square Root Calculation

Solving equations often involves a square root calculation, especially when you have expressions like \( x^2 \) equals a number. Taking the square root is a method to simplify such equations.

• The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). It’s expressed as \( \sqrt{x} \).
• Understanding how to calculate and interpret square roots is essential for solving problems like finding the sides of geometric shapes when their areas are given.
• In our example, solving \( h^2 = 100 \) by taking the square root on both sides gives \( h = \sqrt{100} = 10 \).

Mastering the square root function aids not only in geometry but also enhances comprehension of algebraic manipulations essential in advanced mathematics.