### Video Transcript

Is the length of one side of
the given figure proportional to its area?

So, let’s have a look at the
shape in this question. We can see that there are four
right angles and two sides labeled the same length. So, we must have a square. We’re asked if the length of
one side is proportional to the area. So, let’s recall how to find
the area of a square. And that is that the area of a
square is equal to the length times the length, or the length squared. So, the area of our square is
𝑆 squared.

Let’s recall proportion. If we have two quantities 𝐴
and 𝐵 which are proportional, then that means from one situation to another,
both quantities have been multiplied by the same number. We know that in one situation,
the area of our square is equal to 𝑆 squared. Let’s imagine another situation
where we double the length of our sides. In this case, the area of our
second square, or square two, would be equal to two 𝑆 times two 𝑆, which is
four 𝑆 squared. We can note that the area of
our first square, which we could call square one, was equal to 𝑆 squared. So, the area of square two is
equal to four times the area of square one.

Now, let’s imagine another
situation where we multiply the length of our square by three. So, in this case, the area of
square three would be equal to three 𝑆 times three 𝑆, which is nine 𝑆
squared. And given that our first square
was equal to 𝑆 squared, then this means that the area of square three is equal
to nine times the area of square one. So, if we consider these values
as fractions of the length over the area, in the first situation we have the
length 𝑆 over 𝑆 squared. We then double the lengths, so
the fraction would be two 𝑆 over the area of four 𝑆 squared. And in our final situation, we
had three 𝑆 as the length over the area of nine 𝑆 squared.

These two quantities would be
proportional if we can say that they are multiplied by the same number. However, going from the first
fraction to the second fraction would mean the numerator was multiplied by two
and the denominator was multiplied by four. We can also see that from the
first fraction to the third fraction, we multiplied the numerator by three and
the denominator by nine, which means that these have not been multiplied by the
same number. So, the answer to the question,
is the length of one side of this figure proportional to its area, is no.