Find the value of 𝑥.
We see from the figure that the
lines 𝐴𝐶 and 𝐴𝐵 are transversals that intersect parallel lines 𝐷𝐸 and
𝐵𝐶. And we know that the two pairs
of corresponding angles created by this intersection are equal. That’s angles 𝐷𝐸𝐴 and 𝐵𝐶𝐴
and angles 𝐸𝐷𝐴 and 𝐶𝐵𝐴. This being the case, we can say
the triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐸 are similar triangles since they each have the
common angle 𝐵𝐴𝐶 and their other two angles are also equal.
Now recall that when two
triangles are similar, the ratios of the length of their corresponding sides are
equal. In particular, 𝐴𝐷 is to 𝐴𝐵
as 𝐷𝐸 is to 𝐵𝐶. In other words, 𝐴𝐷 over 𝐴𝐵
is equal to 𝐷𝐸 over 𝐵𝐶. Now, we know that 𝐴𝐷 is equal
to 10 units. 𝐴𝐵 is equal to 10 plus 11
units, that’s 𝐴𝐷 plus 𝐷𝐵. 𝐷𝐸 is equal to 10 units, and
𝐵𝐶 is 𝑥. And so we have 10 over 10 plus
11 is equal to 10 over 𝑥. That is, 10 over 21 is equal to
10 over 𝑥.
Now, solving for 𝑥, we
multiply through by 21𝑥 and divide both sides by 10, and we have 𝑥 equal to
21. Hence, using the given diagram,
we find that 𝑥 is equal to 21 units.