Given that 𝑄𝑅𝑃𝑀 is similar to
𝑆𝑉𝑍𝑊, find the value of 𝑥.
Note that the approximation symbol
here means that the two rectangles are similar. When two polygons are similar, we
know that their corresponding sides are proportional. In this question, we have
corresponding sides 𝑄𝑀 and 𝑆𝑊 along with 𝑀𝑃 and 𝑊𝑍. This means that the ratio of six 𝑥
minus 16 to 13 will be the same as nine 𝑥 minus 33 to 15. Writing this in fractional form
gives us six 𝑥 minus 16 over nine 𝑥 minus 33 is equal to 13 over 15.
In order to calculate 𝑥, we could
cross multiply immediately. However, it is often useful to try
and simplify the fractions first. The numerator and denominator on
the left-hand side can be factored. Six 𝑥 minus 16 is equal to two
multiplied by three 𝑥 minus eight. And nine 𝑥 minus 33 is equal to
three multiplied by three 𝑥 minus 11. The denominators have a common
factor of three, so we can divide both of these by three.
Cross multiplying at this stage
gives us 10 multiplied by three 𝑥 minus eight is equal to 13 multiplied by three 𝑥
minus 11. We get the 10 by multiplying five
by two. Redistributing our parentheses
gives us 30𝑥 minus 80 is equal to 39𝑥 minus 143. Adding 143 to both sides of this
equation gives us 30𝑥 plus 63 is equal to 39𝑥. Subtracting 30𝑥 from both sides
gives us 63 is equal to nine 𝑥. Finally, dividing both sides of
this equation by nine gives us a value of 𝑥 equal to seven.
We can then check this answer by
substituting 𝑥 equals seven into our expressions on the first rectangle. Six multiplied by seven is equal to
42, and subtracting 16 gives us 26. Nine multiplied by seven is equal
to 63. Subtracting 33 from this gives us
30. We can therefore see that our first
rectangle is twice the size of our second rectangle as 26 is double 13 and 30 is
double 15. The scale factor to get from
rectangle 𝑄𝑅𝑃𝑀 to 𝑆𝑉𝑍𝑊 is a half as the corresponding sides of the second
rectangle are half the size of the first one.