Given $\Delta R W S \cong \Delta T U V$ , find the values of $\mathrm{x}$ and $\mathrm{y}$ . Be sure to clearly label each of your variables in your answer. $\newline$ $\begin{array}{l} x= \\ y= \end{array}$

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Precalculus

Solve higher-degree inequalities

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Q. Given

\Delta R W S \cong \Delta T U V

, find the values of

\mathrm{x}

and

\mathrm{y}

. Be sure to clearly label each of your variables in your answer.

\newline

\begin{array}{l} x= \\ y= \end{array}

Identify Angles and Sides: Identify corresponding angles and sides since the triangles are similar.
Set Up Equations: Set up the equations based on the similarity of the triangles. If $RWS$ is similar to $TUV$ , then $\frac{RW}{TU} = \frac{WS}{UV} = \frac{RS}{TV}.$
Write Given Values: Write down the given side lengths and angle measures. Assume $RW = x$ , $WS = y$ , $RS = x + y$ , $TU = 10$ , $UV = 15$ , and $TV = 25$ .
Create Proportion: Create the proportion $\frac{x}{10} = \frac{y}{15} = \frac{x + y}{25}.$
Solve for $x$ and $y$ : Solve the first part of the proportion $\frac{x}{10} = \frac{y}{15}$ . Cross-multiply to get $15x = 10y$ .
Substitute Values: Divide both sides by $5$ to simplify, getting $3x = 2y$ .
Check Solution: Solve for $y$ in terms of $x$ , $y = \frac{3}{2}x$ .
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ .
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ . Substitute $y$ in the equation: $\frac{x}{10} = \frac{x + (\frac{3}{2}x)}{25}$ .
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ . Substitute $y$ in the equation: $\frac{x}{10} = \frac{x + (\frac{3}{2}x)}{25}$ . Simplify the equation: $\frac{x}{10} = \frac{\frac{5}{2}x}{25}$ . Cross-multiply to get $25x = 10\times(\frac{5}{2}x)$ .
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ . Substitute $y$ in the equation: $\frac{x}{10} = \frac{x + (\frac{3}{2}x)}{25}$ . Simplify the equation: $\frac{x}{10} = \frac{\frac{5}{2}x}{25}$ . Cross-multiply to get $25x = 10\times(\frac{5}{2}x)$ . Simplify the equation: $25x = 25x$ . This is always true, so there's no unique solution for $x$ from this equation.
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ . Substitute $y$ in the equation: $\frac{x}{10} = \frac{x + (\frac{3}{2}x)}{25}$ . Simplify the equation: $\frac{x}{10} = \frac{\frac{5}{2}x}{25}$ . Cross-multiply to get $25x = 10\times(\frac{5}{2}x)$ . Simplify the equation: $25x = 25x$ . This is always true, so there's no unique solution for $x$ from this equation. Use the first equation $3x = 2y$ to find $x$ and $y$ . Let's choose a value for $x$ , say $\frac{x}{10} = \frac{x + y}{25}$ $2$ .
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ . Substitute $y$ in the equation: $\frac{x}{10} = \frac{x + (\frac{3}{2}x)}{25}$ . Simplify the equation: $\frac{x}{10} = \frac{\frac{5}{2}x}{25}$ . Cross-multiply to get $25x = 10\times(\frac{5}{2}x)$ . Simplify the equation: $25x = 25x$ . This is always true, so there's no unique solution for $x$ from this equation. Use the first equation $3x = 2y$ to find $x$ and $y$ . Let's choose a value for $x$ , say $\frac{x}{10} = \frac{x + y}{25}$ $2$ . Substitute $\frac{x}{10} = \frac{x + y}{25}$ $2$ into $3x = 2y$ to find $y$ . So, $\frac{x}{10} = \frac{x + y}{25}$ $6$ , which gives $\frac{x}{10} = \frac{x + y}{25}$ $7$ .
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ . Substitute $y$ in the equation: $\frac{x}{10} = \frac{x + (\frac{3}{2}x)}{25}$ . Simplify the equation: $\frac{x}{10} = \frac{(\frac{5}{2})x}{25}$ . Cross-multiply to get $25x = 10*(\frac{5}{2})x$ . Simplify the equation: $25x = 25x$ . This is always true, so there's no unique solution for $x$ from this equation. Use the first equation $3x = 2y$ to find $x$ and $y$ . Let's choose a value for $x$ , say $\frac{x}{10} = \frac{x + y}{25}$ $2$ . Substitute $\frac{x}{10} = \frac{x + y}{25}$ $2$ into $3x = 2y$ to find $y$ . So, $\frac{x}{10} = \frac{x + y}{25}$ $6$ , which gives $\frac{x}{10} = \frac{x + y}{25}$ $7$ . Divide both sides by $\frac{x}{10} = \frac{x + y}{25}$ $8$ to solve for $y$ , getting $y$ $0$ .
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ . Substitute $y$ in the equation: $\frac{x}{10} = \frac{x + (\frac{3}{2}x)}{25}$ . Simplify the equation: $\frac{x}{10} = \frac{(\frac{5}{2})x}{25}$ . Cross-multiply to get $25x = 10*(\frac{5}{2})x$ . Simplify the equation: $25x = 25x$ . This is always true, so there's no unique solution for $x$ from this equation. Use the first equation $3x = 2y$ to find $x$ and $y$ . Let's choose a value for $x$ , say $\frac{x}{10} = \frac{x + y}{25}$ $2$ . Substitute $\frac{x}{10} = \frac{x + y}{25}$ $2$ into $3x = 2y$ to find $y$ . So, $\frac{x}{10} = \frac{x + y}{25}$ $6$ , which gives $\frac{x}{10} = \frac{x + y}{25}$ $7$ . Divide both sides by $\frac{x}{10} = \frac{x + y}{25}$ $8$ to solve for $y$ , getting $y$ $0$ . Check if the values of $\frac{x}{10} = \frac{x + y}{25}$ $2$ and $y$ $0$ satisfy the proportion $y$ $3$ .
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ . Substitute $y$ in the equation: $\frac{x}{10} = \frac{x + (\frac{3}{2}x)}{25}$ . Simplify the equation: $\frac{x}{10} = \frac{\frac{5}{2}x}{25}$ . Cross-multiply to get $25x = 10*(\frac{5}{2})x$ . Simplify the equation: $25x = 25x$ . This is always true, so there's no unique solution for $x$ from this equation. Use the first equation $3x = 2y$ to find $x$ and $y$ . Let's choose a value for $x$ , say $\frac{x}{10} = \frac{x + y}{25}$ $2$ . Substitute $\frac{x}{10} = \frac{x + y}{25}$ $2$ into $3x = 2y$ to find $y$ . So, $\frac{x}{10} = \frac{x + y}{25}$ $6$ , which gives $\frac{x}{10} = \frac{x + y}{25}$ $7$ . Divide both sides by $\frac{x}{10} = \frac{x + y}{25}$ $8$ to solve for $y$ , getting $y$ $0$ . Check if the values of $\frac{x}{10} = \frac{x + y}{25}$ $2$ and $y$ $0$ satisfy the proportion $y$ $3$ . Substitute $x$ and $y$ into the proportion: $y$ $6$ .
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ . Substitute $y$ in the equation: $\frac{x}{10} = \frac{x + (\frac{3}{2}x)}{25}$ . Simplify the equation: $\frac{x}{10} = \frac{(\frac{5}{2})x}{25}$ . Cross-multiply to get $25x = 10\times(\frac{5}{2})x$ . Simplify the equation: $25x = 25x$ . This is always true, so there's no unique solution for $x$ from this equation. Use the first equation $3x = 2y$ to find $x$ and $y$ . Let's choose a value for $x$ , say $\frac{x}{10} = \frac{x + y}{25}$ $2$ . Substitute $\frac{x}{10} = \frac{x + y}{25}$ $2$ into $3x = 2y$ to find $y$ . So, $\frac{x}{10} = \frac{x + y}{25}$ $6$ , which gives $\frac{x}{10} = \frac{x + y}{25}$ $7$ . Divide both sides by $\frac{x}{10} = \frac{x + y}{25}$ $8$ to solve for $y$ , getting $y$ $0$ . Check if the values of $\frac{x}{10} = \frac{x + y}{25}$ $2$ and $y$ $0$ satisfy the proportion $y$ $3$ . Substitute $x$ and $y$ into the proportion: $y$ $6$ . Simplify the proportion: $y$ $7$ , which is true.
State Final Answer: Substitute $y = \frac{3}{2}x$ into the second part of the proportion $\frac{x}{10} = \frac{x + y}{25}$ . Substitute $y$ in the equation: $\frac{x}{10} = \frac{x + (\frac{3}{2}x)}{25}$ . Simplify the equation: $\frac{x}{10} = \frac{\frac{5}{2}x}{25}$ . Cross-multiply to get $25x = 10\times(\frac{5}{2}x)$ . Simplify the equation: $25x = 25x$ . This is always true, so there's no unique solution for $x$ from this equation. Use the first equation $3x = 2y$ to find $x$ and $y$ . Let's choose a value for $x$ , say $\frac{x}{10} = \frac{x + y}{25}$ $2$ . Substitute $\frac{x}{10} = \frac{x + y}{25}$ $2$ into $3x = 2y$ to find $y$ . So, $\frac{x}{10} = \frac{x + y}{25}$ $6$ , which gives $\frac{x}{10} = \frac{x + y}{25}$ $7$ . Divide both sides by $\frac{x}{10} = \frac{x + y}{25}$ $8$ to solve for $y$ , getting $y$ $0$ . Check if the values of $\frac{x}{10} = \frac{x + y}{25}$ $2$ and $y$ $0$ satisfy the proportion $y$ $3$ . Substitute $x$ and $y$ into the proportion: $y$ $6$ . Simplify the proportion: $y$ $7$ , which is true. State the final answer with the values of $x$ and $y$ .