You are told that ΔABC ~ ΔDEC, which means that they are similar triangles. The ratio of similar sides of similar triangles will be the same. This means that the ratio of [tex]\sf\dfrac{BC}{AC}[/tex] should be equal to the ratio of [tex]\sf\dfrac{EC}{DC}[/tex]. BC is equal to 20 + 10 = 30, AC is equal to 'x + 6', EC is equal to 20, and DC is equal to 'x', let's plug these in and solve for 'x':
[tex]\sf\dfrac{BC}{AC}=\dfrac{EC}{DC}[/tex]
[tex]\sf\dfrac{30}{x+6}=\dfrac{20}{x}[/tex]
Cross multiply:
[tex]\sf (x+6)(20)=(30)(x)[/tex]
[tex]\sf 20x+120=30x[/tex]
Subtract 20x to both sides:
[tex]\sf 120=10x[/tex]
Divide 10 to both sides:
[tex]\sf x=\boxed{\sf 12}[/tex]
So DC is 12 units long.
