Question

triangle abc is similar to triangle def. the following ratios of corresponding sides are equal.
\\(\frac{12}{8}=\frac{24}{16}=\frac{2x + 11}{2x+5}\\)
\\(x = \square\\) (type an integer or a simplified fraction.)

Explanation:

Step1: Cross - multiply the proportion

Since $\frac{12}{24}=\frac{8}{16}=\frac{2x + 5}{2x+11}$, we can use the property of equal - ratios. Cross - multiplying the proportion $\frac{2x + 5}{2x+11}=\frac{12}{24}$ (we could also use other equal ratios), we get $24(2x + 5)=12(2x + 11)$.

Step2: Expand both sides

Expand the left - hand side: $24\times2x+24\times5 = 48x+120$. Expand the right - hand side: $12\times2x+12\times11=24x + 132$. So, $48x+120 = 24x+132$.

Step3: Move the terms with $x$ to one side

Subtract $24x$ from both sides: $48x - 24x+120=24x - 24x+132$, which simplifies to $24x+120 = 132$.

Step4: Isolate $x$

Subtract 120 from both sides: $24x+120 - 120=132 - 120$, getting $24x = 12$. Then divide both sides by 24: $x=\frac{12}{24}=\frac{1}{2}$.

Answer:

$\frac{1}{2}$