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The table below shows some properties of regular polygons.



When compared with the independent variable, how many of the remaining three columns represent a linear relationship?

A. 0
B. 1
C. 2
D. 3

Answer: C.2

Step-by-step explanation:

Given: Independent variable= Number of sides

To check how many of the remaining three columns represent a linear relationship, we need to check the rate of change of the three columns with respect to dependent variable.

For number of diagonals

The rate of change of diagonals w.r.t number of sides is constant and that is 1.

For Sum of all interior angles (degrees)

The rate of change of  Sum of all interior angles w.r.t number of sides is constant and that is 180°.

For measure of each angle (degrees)

The rate of change of  measure of each angle w.r.t number of sides is not constant and that is 180°.

Since for vertices 3 , measure of angle=60

vertices 4 , measure of angle=90

From vertices 3 to 4 change in measure of angle is 30

vertices 5 , measure of angle=108

From vertices 4 to 5 change in measure of angle is 18

Thus number of diagonals and Sum of all interior angles are the columns represent a linear relationship when compared with the independent variable.

Therefore there are 2 columns represent a linear relationship when compared with the independent variable.

Solution:

The equation of linear relationship is , y= k x

That is, [tex]\frac{y-value}{x-value}=k({\text{a constant})[/tex]

As, you can see that , ratio of

1.

[tex]\frac{\text{Sum of interior angles of a regular polygon}}{\text{number of sides of that polygon}}={\text{Measure of each interior angle of regular polygon}}[/tex]

2. Also, as number of sides increases, the smallest regular polygon is equilateral triangle(3 sides) , having 0, diagonal, the number of diagonal increases by 1.

→→Regular polygon having sides , n≥3=k, where k is number of diagonals=0,1,2,3,4,[ For example, for, n=3,k=1,n=4, k=2,...]

Column (2)

[tex]C_{4}-C_{3}=C_{3}-C_{2}=C_{2}-C_{1}=1[/tex]

Column (4)

[tex]C_{2}-C_{1}=90-60=30\\\\ C_{3}-C_{2}=108-90=18\\\\ C_{4}-C_{3}=120-108=12[/tex]

Column (3)

[tex]C_{4}-C_{3}=720-540=180\\\\ C_{3}-C_{2}=540-360=180\\\\ C_{2}-C_{1}=360-180=180[/tex]

In, Column (2) and Column(3) the difference between two consecutive values is same, while in column (4) difference between two consecutive values is not same.So, C(2) and C(3) shows linear relationship, column (4) do not.

Option (C) 2 is Right choice.

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