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

Since the triangles are similar, the ratios of corresponding sides are equal. So, $\frac{2x - 8}{x + 6}=\frac{42}{21}$.

Step2: Simplify the right - hand side

Simplify $\frac{42}{21}$ to 2. The equation becomes $\frac{2x - 8}{x + 6}=2$.

Step3: Cross - multiply

Cross - multiplying gives $2x−8 = 2(x + 6)$.

Step4: Expand the right - hand side

Expand $2(x + 6)$ to get $2x−8=2x + 12$.

Step5: Solve for x

Subtract $2x$ from both sides: $2x-2x - 8=2x-2x + 12$, which results in $-8 = 12$, a contradiction. Let's set up the proportion the other way around: $\frac{2x - 8}{42}=\frac{x + 6}{21}$.

Step6: Cross - multiply new proportion

Cross - multiplying gives $21(2x - 8)=42(x + 6)$.

Step7: Expand both sides

Expand: $42x-168 = 42x+252$. Subtracting $42x$ from both sides gives $-168=252$, another contradiction. Let's assume the correct proportion is $\frac{2x - 8}{24}=\frac{42}{x + 6}$.

Step8: Cross - multiply

$(2x - 8)(x + 6)=42\times24$.

Step9: Expand the left - hand side

$2x^2+12x-8x - 48=1008$.

Step10: Simplify the left - hand side

$2x^2 + 4x-48 - 1008=0$, so $2x^2+4x - 1056=0$. Divide through by 2: $x^2+2x - 528=0$.

Step11: Factor the quadratic equation

$(x + 24)(x - 22)=0$.

Step12: Solve for x

Setting each factor equal to zero gives $x+24 = 0$ or $x - 22=0$. So $x=-24$ or $x = 22$. But since lengths cannot be negative in the context of side - lengths of a triangle, we discard $x=-24$.

Answer:

$22$