Question

for the pair of similar triangles, find the appropriate value(s) of x.
2x-4
39
x+6
24
the appropriate value(s) of x is/are □.
(simplify your answer. use a comma to separate answers as needed.)

Explanation:

Step1: Set up proportion for similar triangles

Corresponding sides of similar triangles are proportional. The hypotenuse of the first triangle is 39, hypotenuse of the second is 24; the leg of the first triangle is $2x-4$, corresponding leg of the second is $x+6$. So:
$$\frac{2x-4}{x+6} = \frac{39}{24}$$

Step2: Simplify the fraction

Reduce $\frac{39}{24}$ to lowest terms by dividing numerator and denominator by 3:
$$\frac{2x-4}{x+6} = \frac{13}{8}$$

Step3: Cross-multiply to eliminate fractions

Multiply both sides by $8(x+6)$ to clear denominators:
$$8(2x-4) = 13(x+6)$$

Step4: Expand both sides

Distribute the constants on each side:
$$16x - 32 = 13x + 78$$

Step5: Isolate the variable x

Subtract $13x$ from both sides, then add 32 to both sides:
$$16x - 13x = 78 + 32$$
$$3x = 110$$

Step6: Solve for x

Divide both sides by 3:
$$x = \frac{110}{3}$$

Answer:

$\frac{110}{3}$