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:
$\frac{uv}{-tu}$
$\frac{vu}{rs}$
$\frac{qr}{-tu}$
$\frac{qr}{tu}$

Explanation:

Step1: Recall Slope Formula

The slope of a line is given by \( \text{slope} = \frac{\text{change in } y}{\text{change in } x} \). For a line passing through two points, if we consider the right triangles formed (like \( \triangle QRS \) and \( \triangle TUV \)), the slope can also be represented as \( \frac{\text{vertical change}}{\text{horizontal change}} \) (with sign based on direction).

Step2: Analyze Triangles

• For \( \triangle QRS \), let's say \( QR \) is the vertical side (change in \( y \)) and \( RS \) is the horizontal side (change in \( x \)). But since the line is decreasing (from the figure's appearance, as the triangles are right - angled and the line has a negative slope), the slope should be negative.
• For \( \triangle TUV \), \( UV \) is the vertical side (change in \( y \)) and \( TU \) is the horizontal side (change in \( x \)).
• The slope of line \( QT \) can be calculated using the similar triangles (since \( \triangle QRS \sim \triangle TUV \) as they are right - angled and share the same angle at the intersection of the line and the axes). The slope formula for a line with a negative slope (going down from left to right) is \( \text{slope}=-\frac{\text{opposite side}}{\text{adjacent side}} \) in the right triangle or \( \frac{\text{vertical change (negative)}}{\text{horizontal change (positive)}} \).
• Looking at the options, the slope of \( QT \) should be \( -\frac{QR}{TU} \) (or equivalent from similar triangles). Let's check the options:
• Option 1: \( -\frac{UV}{TU} \): Not matching the triangle sides related to \( \triangle QRS \).
• Option 2: \( \frac{VU}{RS} \): Sign is wrong (should be negative as the line is decreasing) and the sides are not correctly related.
• Option 3: \( -\frac{QR}{TU} \): This matches the slope calculation. The vertical change (from \( Q \) to \( R \) or equivalent in similar triangle) is negative (since the line is going down) and horizontal change is positive, and the ratio of the vertical side of one triangle to the horizontal side of the other (due to similar triangles) gives the slope.
• Option 4: \( \frac{QR}{TU} \): Sign is wrong as the line has a negative slope.

Answer:

\( -\frac{QR}{TU} \) (the third option among the given options, assuming the options are ordered as \( -\frac{UV}{TU} \), \( \frac{VU}{RS} \), \( -\frac{QR}{TU} \), \( \frac{QR}{TU} \))