Question

which formulas represent linear relationships? select all that apply. the area of a triangle whose height and base are the same, ( a=\frac{1}{2} h^{2} ) the perimeter of a regular octagon, ( p = 8s ) the surface area of a cube, ( s a=6 a^{2} ) the perimeter of a semicircle, ( p=pi r + 2r )

Explanation:

Step1: Recall the form of a linear relationship

A linear relationship has the form \(y = mx + b\), where \(m\) and \(b\) are constants and the highest power of the variable is \(1\).

Step2: Analyze \(A=\frac{1}{2}h^{2}\)

The formula \(A = \frac{1}{2}h^{2}\) has a variable \(h\) with power \(2\). So it is a quadratic relationship, not linear.

Step3: Analyze \(p = 8s\)

The formula \(p=8s\) can be written as \(p = 8s+0\), where \(m = 8\) and \(b = 0\). The highest power of the variable \(s\) is \(1\). So it is a linear relationship.

Step4: Analyze \(SA = 6a^{2}\)

The formula \(SA=6a^{2}\) has a variable \(a\) with power \(2\). So it is a quadratic relationship, not linear.

Step5: Analyze \(p=\pi r + 2r\)

Factor out \(r\): \(p=(\pi + 2)r\). This can be written as \(p=(\pi + 2)r+0\), where \(m=\pi + 2\) and \(b = 0\). The highest power of the variable \(r\) is \(1\). So it is a linear relationship.

Answer:

The perimeter of a regular octagon (\(p = 8s\)) and the perimeter of a semicircle (\(p=\pi r + 2r\)) represent linear relationships.