Question

in the figure, $\triangle qrs$ and $\triangle tuv$ are right triangles, $\overline{qt}$ is a straight line. which one of the following is a correct description of the slope of $\overline{qt}$?\
options: $-\dfrac{uv}{tu}$, $\dfrac{vu}{rs}$, $-\dfrac{qr}{tu}$, $\dfrac{qr}{tu}$

Explanation:

Step1: Recall Slope Formula

The slope of a line is given by \( \text{slope} = \frac{\text{rise}}{\text{run}}=\frac{\text{change in } y}{\text{change in } x} \). For right triangles on a line, we can use the legs of the triangles. Also, similar triangles (since \( \triangle QRS \) and \( \triangle TUV \) are right triangles and on the same line, they are similar) have proportional sides. The slope should be consistent for both triangles.

Step2: Analyze Triangle Sides

For \( \triangle QRS \) (right triangle) and \( \triangle TUV \) (right triangle), the slope of \( \overline{QT} \) can be represented as \( \frac{\text{vertical leg}}{\text{horizontal leg}} \) or its negative (depending on direction). Let's check the options:

- Option 1: \( -\frac{UV}{TU} \): If we consider the direction (from \( T \) to \( U \) to \( V \)), but let's see similar triangles. \( \triangle QRS \sim \triangle TUV \) (AA similarity, right angle and same angle at the line). So \( \frac{QR}{RS}=\frac{UV}{TU} \), but slope is \( \frac{\text{rise}}{\text{run}} \). If the line is increasing or decreasing? Wait, the triangles: \( QRS \) and \( TUV \), if \( Q \) is on the left, \( T \) is on the right? Wait, maybe the slope is \( \frac{QR}{RS} \) or \( \frac{UV}{TU} \), but with sign. Wait, the slope formula: if we go from \( Q \) to \( T \), the change in \( y \) and \( x \). Let's assume \( QR \) is vertical (opposite to \( UV \)) and \( RS \) is horizontal (opposite to \( TU \)). So slope \( = \frac{\text{vertical change}}{\text{horizontal change}} \). If the line is going from \( Q \) (lower) to \( T \) (higher), or vice versa? Wait, the option \( \frac{QR}{TU} \): Wait, no. Wait, the correct slope should be \( \frac{QR}{RS} \) (for \( \triangle QRS \)) and since \( \triangle QRS \sim \triangle TUV \), \( \frac{QR}{RS}=\frac{UV}{TU} \). But the sign: if the horizontal change is positive (from \( Q \) to \( T \), moving right) and vertical change is positive (moving up), slope is positive. But if one of the triangles is flipped? Wait, maybe the slope is \( \frac{QR}{TU} \)? Wait, no, let's re-express. Wait, the options: let's check the last option \( \frac{QR}{TU} \)? No, wait the options are:

Wait the options (from the image, the four options are:

1. \( -\frac{UV}{TU} \)

2. \( \frac{VU}{RS} \)

3. \( -\frac{QR}{TU} \)

4. \( \frac{QR}{TU} \)

Wait, maybe I misread. Wait, the problem says "Which one of the following is a correct description of the slope of \( \overline{QT} \)?"

Wait, let's recall that slope is \( \frac{\text{y}_2 - \text{y}_1}{\text{x}_2 - \text{x}_1} \). For points \( Q \) and \( T \), let's say \( Q \) has coordinates \( (x_Q, y_Q) \), \( T \) has \( (x_T, y_T) \). Then \( \triangle QRS \) has legs \( QR = y_Q - y_S \) (vertical) and \( RS = x_S - x_Q \) (horizontal). \( \triangle TUV \) has legs \( UV = y_V - y_U \) and \( TU = x_U - x_T \). Since \( \overline{QT} \) is a straight line, the slope should be \( \frac{QR}{RS} = \frac{UV}{TU} \) (since similar triangles). But the sign: if \( x_T > x_Q \) (moving right) and \( y_T > y_Q \) (moving up), slope is positive. But if one of the triangles is below the line? Wait, maybe the slope is \( \frac{QR}{TU} \)? No, wait \( RS \) and \( TU \): are \( RS \) and \( TU \) equal? No, similar triangles, so \( RS \) and \( TU \) are corresponding horizontal legs, \( QR \) and \( UV \) are corresponding vertical legs. So slope \( = \frac{QR}{RS} = \frac{UV}{TU} \). But the options: the last option is \( \frac{QR}{TU} \)? Wait, no, maybe the horizontal leg is \( TU \) and vertical leg is \( QR \). Wait, maybe I made a mistake. Wait, the correct answer is \( \frac{QR}{TU} \)? No, wait let's think again.

Wait, the slope of a line through two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( \frac{y_2 - y_1}{x_2 - x_1} \). For \( \triangle QRS \), let \( Q = (x_Q, y_Q) \), \( R = (x_R, y_R) \), \( S = (x_S, y_S) \), so \( QR = y_R - y_Q \) (vertical), \( RS = x_S - x_R \) (horizontal). For \( \triangle TUV \), \( T = (x_T, y_T) \), \( U = (x_U, y_U) \), \( V = (x_V, y_V) \), so \( TU = x_U - x_T \) (horizontal), \( UV = y_V - y_U \) (vertical). Since \( \overline{QT} \) is a straight line, the slope from \( Q \) to \( T \) is \( \frac{y_T - y_Q}{x_T - x_Q} \). Notice that \( y_T - y_Q = y_V - y_U = UV \) (if \( R \) and \( U \) are on the same horizontal line? No, maybe \( QR = y_R - y_Q \) and \( UV = y_V - y_U \), and \( x_T - x_Q = x_U - x_T + x_S - x_R \)? No, better to use similar triangles. \( \triangle QRS \sim \triangle TUV \), so \( \frac{QR}{RS} = \frac{UV}{TU} \), which is the slope. But the options: let's check the options again. The fourth option is \( \frac{QR}{TU} \)? No, wait the options (from the image, the four options are:

1. \( -\frac{UV}{TU} \)

2. \( \frac{VU}{RS} \) (wait \( VU \) is \( -UV \), so \( \frac{-UV}{RS} \))

3. \( -\frac{QR}{TU} \)

4. \( \frac{QR}{TU} \)

Wait, maybe the horizontal change is \( TU \) and vertical change is \( QR \). So slope \( = \frac{QR}{TU} \) (if \( TU \) is the horizontal leg, same as \( RS \) in similar triangle). Since \( \triangle QRS \sim \triangle TUV \), \( RS = TU \)? No, similar triangles have proportional sides, not equal. Wait, maybe the problem is that \( RS \) and \( TU \) are the same length? No, the key is that the slope is \( \frac{\text{vertical leg}}{\text{horizontal leg}} \) of either triangle. So for \( \triangle QRS \), slope is \( \frac{QR}{RS} \), and for \( \triangle TUV \), slope is \( \frac{UV}{TU} \). Since they are similar, \( \frac{QR}{RS} = \frac{UV}{TU} \). Now, the sign: if the line is going from \( Q \) to \( T \), moving to the right (positive \( x \)) and up (positive \( y \)), slope is positive. But if one of the triangles is below the line, maybe the vertical leg is negative. Wait, the option \( \frac{QR}{TU} \): if \( RS = TU \) (maybe the horizontal segments are equal), then \( \frac{QR}{RS} = \frac{QR}{TU} \), which would be the slope. So the correct answer is the fourth option, \( \frac{QR}{TU} \).

Answer:

\(\frac{QR}{TU}\) (the last option, assuming the options are labeled as 1 to 4, the fourth option)