Given that 𝐴𝐵𝐶𝐷 is similar to
𝐸𝐹𝐺𝐻, find the values of 𝑥 and 𝑦.
We know that in any similar
polygon, the corresponding sides are proportional. This means that our first step is
to identify the corresponding sides. The side 𝐶𝐷 is corresponding to
𝐺𝐻, 𝐴𝐷 is corresponding to 𝐸𝐻, and 𝐵𝐶 is corresponding to 𝐹𝐺. This means that the ratio of these
three sides must be equal. The ratios 10 to five, eight to two
𝑦 minus 14, and 𝑥 to eight must all be equal. We could set these up in fractional
form to calculate the value of 𝑥 and 𝑦. However, it is clear from the first
ratio that this simplifies to two to one. This means that all the lengths in
the second trapezium will be half the lengths of the first trapezium. The scale factor to go from
trapezium 𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻 is one-half.
We can therefore say that two 𝑦
minus 14 is equal to a half of eight, and a half of eight is four. Adding 14 to both sides of this
equation gives us two 𝑦 is equal to 18. Dividing by two gives us 𝑦 is
equal to nine. We can use the same method to
calculate the value of 𝑥. The length 𝐺𝐻, which is equal to
eight, is half the value of 𝐶𝐷, which is 𝑥. Multiplying both sides of this
equation by two gives us 16 is equal to 𝑥. The missing values are 𝑥 equals 16
and 𝑦 equals nine.