Question

the equation for $overline{qr}$ is $5y=-4x + 41$. is $overline{qr}$ tangent to circle $o$ at $r$? no, because the slope of $overline{or}$ times the slope of $overline{qr}$ does not equal 1. yes, because the slope of $overline{or}$ times the slope of $overline{qr}$ equals 1. no, because the slope of $overline{or}$ times the slope of $overline{qr}$ does not equal - 1. yes, because the slope of $overline{or}$ times the slope of $overline{qr}$ equals - 1.

Explanation:

Step1: Find the slope of $\overline{QR}$

Rewrite the equation $5y=-4x + 41$ in slope - intercept form $y=mx + b$ (where $m$ is the slope). Divide both sides by 5: $y=-\frac{4}{5}x+\frac{41}{5}$. So the slope of $\overline{QR}$, denoted as $m_{QR}=-\frac{4}{5}$.

Step2: Find the slope of $\overline{OR}$

The coordinates of $O(0,0)$ and $R(4,5)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $O(0,0)$ and $R(4,5)$, $m_{OR}=\frac{5 - 0}{4 - 0}=\frac{5}{4}$.

Step3: Check the perpendicularity condition

Two lines are perpendicular if the product of their slopes is - 1. Calculate $m_{OR}\times m_{QR}=\frac{5}{4}\times(-\frac{4}{5})=-1$. If a line is tangent to a circle at a point, the radius at that point is perpendicular to the tangent line.

Answer:

D. Yes, because the slope of $\overline{OR}$ times the slope of $\overline{QR}$ equals - 1.