Question

write an equation that relates the quantities. The area \(A\) of a triangle varies jointly with the product of the lengths of the base \(b\) and the height \(h .\) The constant of proportionality is \(\frac{1}{2}\).

Short answer

The equation is \( A = \frac{1}{2} bh \).

Step-by-step solution

Understand the Problem

The problem states that the area of a triangle varies jointly with the product of the base and height, with a constant of proportionality. The area formula for a triangle is given as a starting point.

Identify the Variables

Let the area be denoted as \(A\), the base as \(b\), and the height as \(h\).

Write the General Proportionality Equation

According to the problem, \(A\) varies jointly with \(b\) and \(h\), so we can write \[ A = k \cdot b \cdot h \] where \(k\) is the constant of proportionality.

Substitute the Constant of Proportionality

The problem states that the constant of proportionality is \frac{1}{2}\, so substitute \ k = \frac{1}{2} \ into the equation. This gives us: \[ A = \frac{1}{2} \cdot b \cdot h \]

Write the Final Equation

Combine the terms to write the final equation that relates the quantities: \[ A = \frac{1}{2} b h \]

Key concepts

headline of the respective core concept

Joint variation occurs when one variable depends on the product of two or more other variables. In this exercise, the area of the triangle \( A \) varies jointly with the base \( b \) and the height \( h \).
Joint variation can be expressed using proportionality equations. Here, this relationship is written as \( A = k \cdot b \cdot h \), where \( k \) is the constant of proportionality.
Understanding joint variation helps us grasp how different geometric properties depend on each other. This builds a foundation for more complex mathematical concepts.

headline of the respective core concept

The proportionality constant is a key part of joint variation equations. It determines how the variables scale together.
In our exercise, the area of the triangle is scaled by the constant of proportionality \( k \,=\, \frac{1}{2} \). Substituting \( k \) in our equation, we get \( A \,=\, \frac{1}{2} b h \).
The value of \( k \) ensures the formula is accurate to real-world measurements. Understanding \( k \) is crucial for correctly applying formulas in various math and science problems.

headline of the respective core concept

Geometry deals with shapes, sizes, and the properties of space.
This exercise focuses on the area of a triangle, a fundamental concept in geometry. The formula \( A \,=\, \frac{1}{2} b h \) comes from how we understand space and shape relationships.
Knowing how geometric formulas are derived helps you use them correctly. It connects abstract mathematical ideas to tangible real-world problems.
Mastering geometry enhances spatial reasoning, critical thinking, and problem-solving skills.