Question

are the area of a square and the length of its side directly proportional quantities? explain.

Explanation:

Define the relationship formula

Using the Proportions knowledge point, we analyze the relationship between the area \(A\) of a square and its side length \(s\). The formula for the area of a square is:

$$A = s^2$$

Analyze direct proportionality

For two quantities to be directly proportional, their ratio must be constant. We check the ratio of area to side length:

$$\frac{A}{s} = \frac{s^2}{s} = s$$

Since this ratio \(s\) changes as the side length changes, it is not a constant value.

Verify with numerical examples

Let \(s_1 = 2\), then \(A_1 = 4\), giving a ratio of \(\frac{4}{2} = 2\).
Let \(s_2 = 3\), then \(A_2 = 9\), giving a ratio of \(\frac{9}{3} = 3\).
Since the ratios are not equal, the quantities are not directly proportional.

Answer:

No, they are not directly proportional because the ratio of the area to the side length is not constant.