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. $\frac{x + 1}{4}=\frac{x + 6}{6}$ 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.

Step2: Identify corresponding sides

The side of length $x + 1$ in $\triangle PQR$ corresponds to the side of length $6$ in $\triangle DFE$, and the side of length $x+3$ in $\triangle PQR$ corresponds to the side of length $x + 6$ in $\triangle DFE$.

Step3: Set up the proportion

The proportion of corresponding sides gives $\frac{x + 1}{6}=\frac{x + 3}{x + 6}$.
Cross - multiply: $(x + 1)(x + 6)=6(x + 3)$.
Expand: $x^{2}+6x+x + 6=6x+18$.
Simplify: $x^{2}+7x + 6=6x + 18$.
Rearrange: $x^{2}+7x-6x+6 - 18=0$.
$x^{2}+x - 12=0$.
Factor: $(x + 4)(x - 3)=0$.
$x=-4$ or $x = 3$. Since length cannot be negative, we consider the proportion based on non - negative lengths.
If we set up the proportion using another pair of corresponding sides, for example, the ratio of the vertical sides and the hypotenuses. The ratio of the vertical sides of the two right - angled similar triangles gives $\frac{4}{x + 6}=\frac{x+3}{x + 8}$.
Cross - multiply: $4(x + 8)=(x + 3)(x + 6)$.
Expand: $4x+32=x^{2}+6x+3x + 18$.
$x^{2}+9x+18-4x - 32=0$.
$x^{2}+5x - 14=0$.
Factor: $(x + 7)(x - 2)=0$.
$x=-7$ or $x = 2$.
The correct proportion based on similar - triangle side - ratio property is $\frac{x + 1}{6}=\frac{x+3}{x + 6}$.
If we consider the ratio of the shorter non - hypotenuse sides and the hypotenuses:
We know that for similar triangles, the ratio of corresponding sides is equal. So, $\frac{x + 1}{6}=\frac{x+3}{x + 6}$.
In terms of the given options, when we use the property of similar triangles and set up the ratio of corresponding sides, we get $\frac{x + 1}{6}=\frac{x+3}{x + 6}$.

Answer:

C. $\frac{x + 1}{6}=\frac{x+3}{x + 6}$