Question

△ dns ~△ arh. what is the value of x? enter your answer in the box. □ units

Explanation:

Step1: Identify similar triangles ratio

Since $\triangle DNS \sim \triangle ARH$, corresponding sides are proportional. Let's assume the sides: $DN = 77$, $NS = 119$, and the corresponding side to $AR$ (length $44$) is $DN$, and the side with $x$ (let's say $RH$) corresponds to $NS$. Wait, maybe better: Let the ratio of similarity be $\frac{AR}{DN}=\frac{44}{77}=\frac{4}{7}$. Then, the side corresponding to $NS$ (length $119$) in the smaller triangle is $x$, so $\frac{x}{119}=\frac{4}{7}$.

Step2: Solve for x

Cross - multiply: $7x = 119\times4$. Calculate $119\times4 = 476$. Then $x=\frac{476}{7}=68$. Wait, maybe I mixed up. Wait, maybe the sides: $DN = 77$, $AR = 44$, $NS = 119$, and $RH = x$. So $\frac{AR}{DN}=\frac{RH}{NS}$. So $\frac{44}{77}=\frac{x}{119}$. Simplify $\frac{44}{77}=\frac{4}{7}$. Then $\frac{4}{7}=\frac{x}{119}$. Cross - multiply: $7x = 4\times119 = 476$. Then $x=\frac{476}{7}=68$. Wait, but maybe the other way. Wait, maybe $DN$ corresponds to $AR$, and $NS$ corresponds to $RH$. So ratio is $\frac{AR}{DN}=\frac{44}{77}=\frac{4}{7}$. Then $RH=\frac{4}{7}\times NS=\frac{4}{7}\times119 = 68$.

Answer:

68