Question

for the pair of similar triangles, find the appropriate value(s) of x.
the appropriate value(s) of x is/are .
(simplify your answer. use a comma to separate answers as needed.)

Explanation:

Step1: Set up proportion of sides

Since the triangles are similar, corresponding sides are proportional:
$\frac{2x-4}{24} = \frac{39}{x+6}$

Step2: Cross-multiply to eliminate fractions

$(2x-4)(x+6) = 24 \times 39$
Expand left side and calculate right side:
$2x^2 + 12x - 4x - 24 = 936$
$2x^2 + 8x - 24 = 936$

Step3: Simplify quadratic equation

Subtract 936 from both sides:
$2x^2 + 8x - 960 = 0$
Divide entire equation by 2:
$x^2 + 4x - 480 = 0$

Step4: Factor the quadratic equation

Find two numbers that multiply to -480 and add to 4: 24 and -20.
$(x + 24)(x - 20) = 0$

Step5: Solve for x and validate

Set each factor equal to 0:
$x + 24 = 0 \implies x = -24$
$x - 20 = 0 \implies x = 20$
Since side lengths cannot be negative, discard $x=-24$ (substituting $x=-24$ gives negative side lengths for both triangles).

Answer:

20