Question

If the base of a triangle is 8 inches and the area of the triangle is 48 square inches, what is the height of the triangle? (A) 6 inches (B) 8 inches (C) 10 inches (D) 12 inches

Short answer

The height of the triangle is 12 inches (D).

Step-by-step solution

Write down the given information

The base of the triangle (b) is 8 inches, and the area (A) is 48 square inches.

Write down the formula for the area of a triangle

The formula for the area of a triangle is A = (1/2)bh.

Substitute the given values into the formula

We will now substitute the given values, A = 48 and b = 8, into the formula: 48 = (1/2)(8)h

Solve for height (h)

To find the height, we will solve the equation for h: 48 = (1/2)(8)h 48 = 4h Divide both sides by 4 to solve for h: h = 12 The height of the triangle is 12 inches, which corresponds to the answer choice (D).

Key concepts

Geometry

Geometry is a branch of mathematics focused on the study of shapes, their properties, and spatial relationships. Triangles are one of the basic building blocks in geometry. Understanding the properties of triangles is crucial for solving many geometric problems.
For any triangle, it consists of three sides and three angles. Triangles can be classified based on their side lengths (equilateral, isosceles, and scalene) or their angles (acute, obtuse, and right triangles). Each type of triangle has unique properties, which are useful in various applications.
The area of a triangle is a fundamental property that helps us understand its size and the space it occupies. To find the area of a triangle, one of the commonly used formulas is: \( A = \frac{1}{2}bh \), where \( b \) is the base and \( h \) is the height perpendicular to the base. This relationship highlights the interdependence between a triangle's dimensions and its area.

Mathematics

Mathematics provides the tools and frameworks for solving problems involving quantities, shapes, and relationships. In the context of the triangle problem, we use algebra to manipulate a formula to find an unknown value—here, the height. The formula for the area of a triangle, \( A = \frac{1}{2}bh \), demonstrates a basic algebraic relationship where you can determine one quantity if the other two are known.
Substitution is a key mathematical technique used here. We substitute the known values (48 for the area and 8 for the base) into the formula to transform it into an equation that can be solved for the unknown height.
This simple algebraic operation—solving an equation—is an integral part of mathematics. It involves rearranging the components of the equation using inverse operations such as division and multiplication, making it an efficient way to find missing elements in geometric problems.

Problem Solving

Problem-solving is a systematic process of finding solutions to complex or challenging situations. In mathematical problems like our triangle exercise, this involves a series of steps. The goal is to isolate the unknown variable and determine its value.
Here's how effective problem-solving is applied:

• First, identify and write down the known information. This gives a clear basis for solving the problem.
• Next, apply a relevant formula. In our exercise, the area formula for triangles connects the base, height, and area.
• Substitute the known values into the formula to form an equation, as this translates the word problem into a mathematical problem.
• Finally, solve the equation by isolating the desired variable, using mathematical operations like division, multiplication, or both.

This structured approach not only provides a clear path to the solution but also develops logical and analytical thinking. It's useful beyond geometry, applicable in various real-world problems where logical sequences and reasoning are required.