Question

in the figure shown, △pqr~△dfe. 7. which equation can be used to find the value of x? a. x + 1 = 6 b. x + 3 = 6 c. $\frac{x + 1}{4}=\frac{x + 6}{x + 8}$ d. $\frac{x + 1}{x + 3}=\frac{x + 6}{x + 8}$

Explanation:

Step1: Recall similarity - side - ratio property

For similar triangles \(\triangle PQR\sim\triangle DFE\), the ratios of corresponding sides are equal.
The ratio of the side with length \(x + 1\) in \(\triangle PQR\) to the side with length \(x+6\) in \(\triangle DFE\) is equal to the ratio of the side with length \(x + 3\) in \(\triangle PQR\) to the side with length \(x + 8\) in \(\triangle DFE\).

Step2: Set up the proportion

We get the proportion \(\frac{x + 1}{x+6}=\frac{x + 3}{x + 8}\).
Cross - multiplying gives \((x + 1)(x + 8)=(x + 3)(x + 6)\).
Expanding both sides: \(x^{2}+8x+x + 8=x^{2}+6x+3x + 18\).
Simplifying, \(x^{2}+9x + 8=x^{2}+9x + 18\) (this is wrong).
The correct proportion based on corresponding sides is \(\frac{x+1}{6}=\frac{x + 3}{x + 8}\).
Cross - multiplying: \((x + 1)(x + 8)=6(x + 3)\).
Another correct way is to use the ratio \(\frac{x+1}{6}=\frac{x + 3}{x + 8}\) or \(\frac{x+1}{x + 6}=\frac{4}{x + 8}\).
If we consider the ratio of corresponding non - hypotenuse sides, we have \(\frac{x+1}{6}=\frac{4}{x + 8}\).
Cross - multiplying gives \((x + 1)(x + 8)=24\).
The proportion based on the sides of similar triangles is \(\frac{x+1}{6}=\frac{x + 3}{x + 8}\), which can be rewritten as \(\frac{x + 1}{6}=\frac{x+3}{x + 8}\Rightarrow(x + 1)(x + 8)=6(x + 3)\).
The correct proportion for the given similar triangles is \(\frac{x+1}{6}=\frac{x + 3}{x + 8}\), and cross - multiplying gives us a quadratic equation to solve for \(x\).
The proportion based on similar - triangle side ratios is \(\frac{x+1}{6}=\frac{x + 3}{x + 8}\).

Answer:

C. \(\frac{x + 1}{6}=\frac{x + 3}{x + 8}\)