Question

the equation for \\(\overline{qr}\\) is \\(5y = -4x + 41\\). is \\(\overline{qr}\\) tangent to circle \\(o\\) at \\(r\\)?
\\(r (4, 5)\\)
no, because the slope of \\(\overline{or}\\) times the slope of \\(\overline{qr}\\) does not equal \\(-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\\).
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 \( 5y = -4x + 41 \), solve for \( y \): \( y = -\frac{4}{5}x + \frac{41}{5} \). Slope \( m_{QR} = -\frac{4}{5} \).

Step3: Check product of slopes

Multiply \( m_{OR} \) and \( m_{QR} \): \( \frac{5}{4} \times (-\frac{4}{5}) = -1 \). Perpendicular lines (product -1) mean \( \overline{QR} \) is tangent (radius \( \overline{OR} \) is perpendicular to tangent at point of contact).

Answer:

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