Given that the rectangles shown are
similar, what is 𝑥?
We’re told in the question that
these two rectangles are mathematically similar, which means that two things are
true. Firstly, all pairs of corresponding
angles between the two rectangles are congruent. Now, for a pair of rectangles, this
is true even for nonsimilar rectangles as all the interior angles in a rectangle are
90 degrees. Secondly, and more importantly
here, all pairs of corresponding sides are in proportion.
We’re looking to find the value of
𝑥 which represents a side length in the larger rectangle. We therefore need to know the scale
factor 𝑆 that is the multiplier that takes us from the smaller rectangle to the
larger. We can use any pair of
corresponding sides to work out the scale factor. It’s equal to the new length
divided by the original length. From the figure, we can see that we
have a pair of corresponding sides of 29 centimeters on the smaller rectangle and 58
centimeters on the larger. We therefore divide the new length
of 58 by the original length of 29, giving a scale factor of 58 over 29, which
simplifies to two.
Remember, the scale factor is
always a multiplier. So this tells us that the lengths
on the larger rectangle are all twice the corresponding lengths on the smaller
rectangle. To work out the value of 𝑥 then,
we need to take the corresponding length on the smaller rectangle, 26, and multiply
it by the scale factor of two, which gives 52. So we found the value of 𝑥. If we had wanted to calculate a
length on the smaller rectangle rather than one on the larger rectangle, we could’ve
used the scale factor of one-half. That’s the reciprocal of two. This could’ve been found by
dividing the length of 29 centimeters by the corresponding length of 58
We can check our value for 𝑥 by
multiplying it by one-half, so 52 multiplied by one-half or half of 52, which is
indeed equal to 26. In formulae, we may think of this
as dividing by two. But remember, the scale factor is
always a multiplier, so we’re using a multiplier of one-half.