Question

the equation for $overline{qr}$ is $5y=-4x + 41$. is $overline{qr}$ tangent to circle $o$ at $r$? r(4, 5)

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 $\overline{QR}$, $m_{QR}=-\frac{4}{5}$.

Step2: Recall the property of a tangent to a circle

A line is tangent to a circle at a point if and only if the radius at that point is perpendicular to the tangent line. For two perpendicular lines with slopes $m_1$ and $m_2$, $m_1\times m_2=- 1$.

Step3: Determine the condition for tangency

If $\overline{QR}$ is tangent to circle $O$ at $R$, the slope of the radius $\overline{OR}$ times the slope of $\overline{QR}$ should equal - 1.

Answer:

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