Question

the equation for (overline{qr}) is (5y = -4x + 41). is (overline{qr}) tangent to circle (o) at (r)? image: circle (o) centered at the origin, point (r(4, 5)) on the circle, line (overline{qr}) through (q) (on (y)-axis) and (r). options: 1. no, because the slope of (overline{or}) times the slope of (overline{qr}) does not equal 1. 2. yes, because the slope of (overline{or}) times the slope of (overline{qr}) equals 1. 3. no, because the slope of (overline{or}) times the slope of (overline{qr}) does not equal (-1). 4. yes, because the slope of (overline{or}) times the slope of (overline{qr}) equals (-1).

Explanation:

Step1: Find slope of \( \overline{OR} \)

Points \( O(0,0) \) and \( R(4,5) \). Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). So \( m_{OR} = \frac{5 - 0}{4 - 0} = \frac{5}{4} \).

Step2: Find slope of \( \overline{QR} \)

Equation \( 5y = -4x + 41 \), rewrite as \( y = -\frac{4}{5}x + \frac{41}{5} \). Slope \( m_{QR} = -\frac{4}{5} \).

Step3: Multiply slopes

\( m_{OR} \times m_{QR} = \frac{5}{4} \times (-\frac{4}{5}) = -1 \). Perpendicular lines (slopes product -1) imply tangent (radius \( OR \) is perpendicular to tangent \( QR \) at \( R \)).

Answer:

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