Question

1. $\triangle qrs \sim \triangle tuv$

Explanation:

Step1: Identify corresponding sides

Since \(\triangle QRS \sim \triangle TUV\), the corresponding sides are proportional. So, \(\frac{RS}{UV}=\frac{RQ}{UT}\).
Given \(RS = 54\), \(UV = 36\), \(RQ = 24\), and \(UT=x + 5\).

Step2: Set up the proportion

Substitute the values into the proportion: \(\frac{54}{36}=\frac{24}{x + 5}\).

Step3: Cross - multiply

Cross - multiplying gives \(54(x + 5)=36\times24\).

Step4: Simplify the right - hand side

Calculate \(36\times24 = 864\), so the equation becomes \(54(x + 5)=864\).

Step5: Divide both sides by 54

Divide both sides of the equation by 54: \(x + 5=\frac{864}{54}\).

Step6: Simplify the right - hand side

Simplify \(\frac{864}{54}=16\), so \(x + 5 = 16\).

Step7: Solve for x

Subtract 5 from both sides: \(x=16 - 5\).

Answer:

\(x = 11\)