Question

1. δnqp ~ δcde
2. δrst ~ δref

Explanation:

For the first similar triangles problem ($\triangle NQP \sim \triangle CDE$):

Step1: Set up proportion

Since the triangles are similar, corresponding sides are proportional. So $\frac{x}{6} = \frac{77}{11}$.

Step2: Cross - multiply

Cross - multiplying gives $11x = 6\times77$. Calculate $6\times77 = 462$, so $11x = 462$.

Step3: Solve for x

Divide both sides by 11: $x=\frac{462}{11}=42$.

For the second similar triangles problem ($\triangle RST \sim \triangle REF$):

Step1: Set up proportion

For similar triangles $\triangle RST$ and $\triangle REF$, the ratio of corresponding sides should be equal. So $\frac{60}{21}=\frac{35}{12x}$ (assuming the sides are corresponding as per the diagram, we can also set it as $\frac{60}{12x}=\frac{35}{21}$ which is equivalent). Let's use $\frac{60}{21}=\frac{35}{12x}$.

Step2: Cross - multiply

Cross - multiplying gives $60\times12x = 35\times21$. Calculate $35\times21 = 735$ and $60\times12x=720x$. So $720x = 735$.

Step3: Solve for x

Divide both sides by 720: $x=\frac{735}{720}=\frac{49}{48}\approx1.02$ (if we use the other proportion $\frac{60}{12x}=\frac{35}{21}$:

Step1: Cross - multiply

$60\times21 = 35\times12x$

Step2: Calculate

$1260 = 420x$

Step3: Solve for x

$x = \frac{1260}{420}=3$ (this seems more likely if the sides are $60$ and $12x$ corresponding to $35$ and $21$ in a different order. Let's re - examine the diagram. If $RT = 60$, $RF=12x$, $RS = 35$, and $RE = 21$, then the correct proportion is $\frac{RT}{RF}=\frac{RS}{RE}$, so $\frac{60}{12x}=\frac{35}{21}$)

Step1: Cross - multiply

$60\times21=35\times12x$

Step2: Simplify

$1260 = 420x$

Step3: Solve for x

$x=\frac{1260}{420} = 3$

Answer:

s:
For the first problem, $x = 42$.
For the second problem (correcting the proportion based on likely corresponding sides), $x = 3$.