Question

in the figure, δqrs and δtuv are right triangles, qt is a straight line. which one of the following is a correct description of the slope of qt? options: -uv/tu, vu/rs, -qr/tu, qr/tu

Explanation:

Step1: Recall Slope Formula

Slope of a line is $\frac{\text{change in } y}{\text{change in } x}$, and for a line with negative slope (since QT is decreasing), it's negative. For right triangles, slope can be related to the legs.

Step2: Analyze Triangles

$\triangle QRS$ and $\triangle TUV$ are right triangles. For line QT, the "rise over run" (slope) will use the vertical and horizontal changes. In $\triangle TUV$, vertical leg is UV (but since it's downward, negative) and horizontal leg is TU. So slope could be $-\frac{UV}{TU}$. Also, $\triangle QRS$: vertical leg QR (downward, negative) and horizontal leg RS? Wait, no—QT is a straight line, so the slope should be consistent. Let's check the options. Option 1: $-\frac{UV}{TU}$—since UV is vertical (opposite to rise direction) and TU is horizontal, so slope is rise/run = -UV/TU (because rise is negative if going down). Let's verify: in right triangle, slope is (vertical change)/(horizontal change). For line QT, moving from T to Q (or Q to T), vertical change is -UV (if UV is length, direction is down) and horizontal change is TU (right). So slope = -UV/TU. Also, in $\triangle QRS$, vertical change is -QR (down) and horizontal change is RS? Wait, no, maybe the triangles are similar. Wait, the key is that the slope of QT is equal to the slope calculated from either triangle. So for $\triangle TUV$, slope is ( - UV ) / TU = -UV/TU. Let's check the options: first option is $-\frac{UV}{TU}$, which matches. Let's see other options: VU/RS—VU is same as UV, RS is horizontal for QRS, not TU. -QR/TU—QR is from QRS, TU is from TUV, not same horizontal. QR/TU—positive, but slope is negative. So correct is first option: $-\frac{UV}{TU}$.

Answer:

A. $-\dfrac{UV}{TU}$ (assuming the first purple box is option A, with text "$-\dfrac{UV}{TU}$")