Question

1. $\triangle qrs \sim \triangle tuv$; find $x$

Explanation:

Step1: Identify corresponding sides

Since $\triangle QRS \sim \triangle TUV$, the ratios of corresponding sides are equal. So, $\frac{QR}{TU}=\frac{RS}{UV}$. Here, $QR = 24$, $TU = x + 5$, $RS = 54$, $UV = 36$.

Step2: Set up the proportion

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

Step3: Cross - multiply

Cross - multiplying gives $54(x + 5)=24\times36$

Step4: Simplify the right - hand side

Calculate $24\times36 = 864$, so the equation becomes $54(x + 5)=864$

Step5: Divide both sides by 54

$\frac{54(x + 5)}{54}=\frac{864}{54}$, which simplifies to $x + 5 = 16$

Step6: Solve for x

Subtract 5 from both sides: $x=16 - 5=11$

Answer:

$x = 11$