what is the length of $overline{dc}$?\n2 units\n3 units\n6 units\n9 units

Answer

Explanation:

Step1: Identify similar triangles

Since $\angle AED=\angle ABC$ and $\angle A$ is common to both $\triangle AED$ and $\triangle ABC$, $\triangle AED\sim\triangle ABC$ by the AA (angle - angle) similarity criterion.

Step2: Set up the proportion

For similar triangles, the ratios of corresponding sides are equal. We know that $\frac{AE}{AB}=\frac{AD}{AC}$. Given $AE = 12$, $EB=4$, so $AB=AE + EB=12 + 4=16$, and $AD = 9$. Let $DC=x$, then $AC=AD + DC=9 + x$. The proportion becomes $\frac{12}{16}=\frac{9}{9 + x}$.

Step3: Cross - multiply

Cross - multiplying the proportion $\frac{12}{16}=\frac{9}{9 + x}$ gives us $12(9 + x)=16\times9$. Expanding the left - hand side: $108+12x = 144$.

Step4: Solve for $x$

Subtract 108 from both sides: $12x=144 - 108=36$. Divide both sides by 12: $x = 3$.

Answer:

3 units