Find the value of 𝑥.
When we look at our diagram, we see
a larger triangle that is cut by the segment 𝐷𝐸. And this line segment 𝐷𝐸 is
parallel to one of the side lengths 𝐵𝐶. And when a triangle is cut by a
line segment that’s parallel to one of its side lengths, two similar triangles are
created. So we can say that if we’re given
line segment 𝐷𝐸 is parallel to 𝐵𝐶, the smaller triangle, triangle 𝐴𝐷𝐸, is
similar to the larger triangle, triangle 𝐴𝐵𝐶. So we can say that 𝐴𝐷 over 𝐴𝐵
is equal to 𝐷𝐸 over 𝐵𝐶. This is because in similar
triangles corresponding side lengths are proportional.
Side length 𝐴𝐷 measures 10, but
what about 𝐴𝐵? And this is where we need to be
very careful. Side length 𝐴𝐵 is not equal to
11. Side length 𝐴𝐵 is the full
distance from vertex 𝐴 to vertex 𝐵, which is 21. So 𝐴𝐷 over 𝐴𝐵 will be equal to
10 over 21. We know that side length 𝐷𝐸 is
equal to 10 as well. So we now have a proportion that
says 10 over 21 is equal to 10 over 𝐵𝐶. And that means 𝐵𝐶 must be equal
to 21 so that these side lengths stay in proportion. Since side length 𝐵𝐶 equals 21,
our missing 𝑥-value is 21.