Question

complete these steps in geogebra. create a circle that is centered at point a(5,2) and goes through point b(7,6). use the segment tool to draw radius \\(\overline{ab}\\). draw the line that is tangent to circle a and goes through point b. use the slope tool to determine the slopes of radius \\(\overline{ab}\\) and the tangent line that it intersects. type the correct answer in each box. use numerals instead of words. the slope of radius \\(\overline{ab}\\) is \\(\square\\). the slope of the tangent line is \\(\square\\).

Explanation:

Step1: Calculate slope of radius AB

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points A(5, 2) and B(7, 6), $x_1 = 5, y_1 = 2, x_2 = 7, y_2 = 6$.
So slope of AB is $\frac{6 - 2}{7 - 5} = \frac{4}{2} = 2$.

Step2: Determine slope of tangent line

The tangent to a circle is perpendicular to the radius at the point of contact. The product of slopes of perpendicular lines is -1. Let the slope of tangent be $m_t$. Then $2 \times m_t = -1$, so $m_t = -\frac{1}{2}$.

Answer:

The slope of radius $\overline{AB}$ is $\boldsymbol{2}$.
The slope of the tangent line is $\boldsymbol{-\frac{1}{2}}$.