Problem 25 - Algebra 2

Question

Use the equation \(d=180(c-2)\) that gives the total number of degrees \(d\) in any convex polygon with \(c\) sides. Write this equation in slope-intercept form.

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = 180x - 360\).

Step 1: Identify the Equation

The given equation is \(d = 180(c-2)\). This represents the relationship between the number of sides \(c\) and the total degrees \(d\) in a convex polygon.

Step 2: Distribute 180 to the Terms Inside Parentheses

Distribute the 180 to both \(c\) and \(-2\) in the equation: \[d = 180 imes c - 180 imes 2\]This simplifies to:\[d = 180c - 360\]

Step 3: Identify the Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this context, you can equate \(d\) to \(y\) and \(c\) to \(x\). Thus, the equation \(d = 180c - 360\) is already in the form \(y = mx + b\).

Step 4: Write the Equation in Slope-Intercept Form

Substitute \(d\) and \(c\) into the slope-intercept form expression: \[y = 180x - 360\]Hence, the equation in slope-intercept form is:\(y = 180x - 360\)

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations so that they are easy to read and interpret. It is expressed as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) represents the y-intercept.

• Slope \(m\): This is a measure of the steepness of the line. When the slope is positive, the line rises as it moves from left to right. Conversely, a negative slope means the line falls.
• Y-intercept \(b\): This is the point where the line crosses the y-axis. It is the value of \(y\) when \(x = 0\).

Understanding and recognizing the slope and y-intercept in an equation not only helps in graphing the equation but also in interpreting the relationship it describes. In any real-world application, such as calculating angles in a polygon, identifying these components makes it easier to manipulate and solve problems involving linear patterns.

Polygons

Polygons are two-dimensional shapes with straight sides. They can have any number of sides greater than two. Common polygons include triangles, quadrilaterals, pentagons, and hexagons.

• Convex Polygons: In such polygons, all the interior angles are less than 180 degrees, and the vertices point outwards.
• Concave Polygons: If a polygon has at least one interior angle greater than 180 degrees, or if some vertices point inward, it is concave.

Each convex polygon follows a precise relationship between the number of its sides \(c\) and the total sum of its internal angles \(d\). The formula \(d = 180(c-2)\) is derived from the concept that a convex polygon can be divided into \(c-2\) triangles, each contributing 180 degrees to the total sum of internal angles.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operational symbols. They are used to represent real-world situations and to solve mathematical problems.

• Variables: Symbols like \(x\) or \(c\) that represent unknown values.
• Constants: These are fixed numbers in expressions, like 180 in the context of polygons.
• Operations: Include addition, subtraction, multiplication, and division. In our example, 180 is multiplied by \(c-2\) in the expression \(d = 180(c-2)\).

Algebraic expressions are fundamental in forming equations and functions that describe how different quantities relate to each other. When we transform an expression like \(d = 180(c-2)\) into an equation in slope-intercept form, we clarify how changes in one quantity (i.e., number of sides) affect another (i.e., total degrees).