Question

find the value of x that makes $\triangle def \sim \triangle xyz$.

Explanation:

Step1: Set up proportion of sides

Since $\triangle DEF \sim \triangle XYZ$, corresponding sides are proportional:
$\frac{DE}{XY} = \frac{DF}{XZ}$
Substitute values: $\frac{10}{x-1} = \frac{3(x-1)}{7.5}$

Step2: Cross-multiply to eliminate fractions

$10 \times 7.5 = 3(x-1)^2$
$75 = 3(x-1)^2$

Step3: Isolate the squared term

Divide both sides by 3:
$\frac{75}{3} = (x-1)^2$
$25 = (x-1)^2$

Step4: Solve for $x-1$

Take square root of both sides:
$x-1 = 5$ (we discard negative root as side length is positive)

Step5: Solve for $x$

Add 1 to both sides:
$x = 5 + 1$

Answer:

$6$