Question

find the value of x. 6.

Explanation:

Step1: Identify similar triangles

Triangles ABE and ACD are similar because they have two equal angles ($\angle A$ is common and $\angle AEB=\angle ACD = 90^{\circ}$).

Step2: Set up proportion

For similar triangles, the ratios of corresponding sides are equal. So, $\frac{AE}{AC}=\frac{AB}{AD}$. We know $AE = 8$, $AB = 12$, $AC=12 + BC$, and $AD=8 + x$. Also, from the right - triangle with angle $25^{\circ}$ and side lengths, we can use the property of similar - triangles. The ratio of the sides of similar triangles gives $\frac{AE}{AC}=\frac{AB}{AD}$. Substituting the known values, we have $\frac{8}{8 + x}=\frac{12}{12 + 18}$.

Step3: Cross - multiply

Cross - multiplying the proportion $\frac{8}{8 + x}=\frac{12}{30}$ gives $12\times(8 + x)=8\times30$.
Expanding the left - hand side: $96+12x = 240$.

Step4: Solve for x

Subtract 96 from both sides: $12x=240 - 96$.
$12x = 144$.
Divide both sides by 12: $x=\frac{144}{12}=12$.

Answer:

12