Given that the two polygons are
similar, find the value of 𝑥.
We know that any similar polygons
have corresponding angles that are congruent and corresponding sides that are
proportional. Due to the orientation of these
shapes, it may not immediately be obvious which sides are corresponding. In order to work this out, it is
useful to identify the corresponding angles first. One pair of corresponding sides are
𝐵𝐶 and 𝑇𝐻. A second pair of corresponding
sides are therefore 𝐶𝐷 and 𝐻𝐽.
As the corresponding sides are
proportional, we know that the ratios two to six and four 𝑥 minus 37 to two 𝑥
minus 11 must be equal. Writing this in fractional form, we
have two over four 𝑥 minus 37 is equal to six over two 𝑥 minus 11. Both of the numerators here are
divisible by two. We can then cross multiply to give
us one multiplied by two 𝑥 minus 11 is equal to three multiplied by four 𝑥 minus
Distributing our parentheses gives
us two 𝑥 minus 11 is equal to 12𝑥 minus 111. Adding 111 to both sides of this
equation gives us two 𝑥 plus 100 is equal to 12𝑥. We can then subtract two 𝑥 from
both sides of this equation, which gives us 100 is equal to 10𝑥. Finally, dividing both sides of
this equation by 10 gives us a value of 𝑥 equal to 10.
We can then substitute this value
back into the expressions for the lengths of 𝐵𝐶 and 𝑇𝐻 to check our answer. Four multiplied by 10 is equal to
40. Subtracting 37 from this gives us
three. Two multiplied by 10 is equal to
20, and subtracting 11 from this gives us nine. The ratios two to six and three to
nine are identical as they can both be simplified to one to three.
An alternative method in this
question would be to initially recognize that the scale factor was three. This is because the length 𝐽𝐻 is
three times the length of 𝐶𝐷. We could then have set up the
equation two 𝑥 minus 11 is equal to three multiplied by four 𝑥 minus 37 as the
length 𝑇𝐻 is three times the length of 𝐵𝐶. Following this method would also
have got us a value of 𝑥 equal to 10.