Question

i \can\ similarity δabc ~ δdfe. find the value of x. a d 2x + 2 f 40° x + 3 c 40° b 16 e 24 a 3 c 7 b 14 d 5 14 o a

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle ABC\sim\triangle DFE\), the ratios of corresponding sides are equal. That is \(\frac{AB}{DF}=\frac{BC}{FE}\).
Here, \(AB = 2x + 2\), \(DF=x + 3\), \(BC = 24\), and \(FE = 16\). So we have the equation \(\frac{2x+2}{x + 3}=\frac{24}{16}\).

Step2: Simplify the equation

Simplify \(\frac{24}{16}\) to \(\frac{3}{2}\). The equation becomes \(\frac{2x+2}{x + 3}=\frac{3}{2}\).
Cross - multiply: \(2(2x + 2)=3(x + 3)\).
Expand: \(4x+4 = 3x+9\).

Step3: Solve for \(x\)

Subtract \(3x\) from both sides: \(4x-3x+4=3x - 3x+9\), which gives \(x+4 = 9\).
Subtract \(4\) from both sides: \(x=9 - 4\).

Answer:

\(x = 5\), so the answer is D.