Question

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

options:
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 \(\overline{QR}\)

Given the equation \(5y=-4x + 41\), rewrite it 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: Find the slope of \(\overline{OR}\)

The center \(O=(0,0)\) and the point \(R=(4,5)\). Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), for \(O(0,0)\) and \(R(4,5)\), \(m_{OR}=\frac{5 - 0}{4 - 0}=\frac{5}{4}\).

Step3: Check the product of the slopes

Multiply the slopes: \(m_{OR}\times m_{QR}=\frac{5}{4}\times(-\frac{4}{5})=- 1\).
A line is tangent to a circle at a point if and only if the radius to that point is perpendicular to the tangent line. Two lines with slopes \(m_1\) and \(m_2\) are perpendicular if \(m_1\times m_2=-1\).

Answer:

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