Question

radius of a circle and tangent lines
this activity will help you meet these educational goals:

• 21st century skills—you will use critical - thinking and problem - solving skills.

direction
read the instructions for this self - checked activity. type in your response to each question, and check your answers. at the end of the activity, write a brief evaluation of your work.
activity
you will use the geogebra geometry tool to explore the relationship between the radius of a circle and a tangent line that intersects the radius. go to relationships in circles icon and complete each step below. if you need help, follow these instructions icon for using geogebra
question 1
create a circle with center a and a radius of your choice. create a point b on the circle, and find the coordinates of b. draw the radius $overline{ab}$. what is the slope - intercept form ($y = mx + b$) of the equation of $overleftrightarrow{ab}$? show your work.

Explanation:

Step1: Assume center coordinates

Let the center of the circle $A$ have coordinates $(x_1,y_1)$ and the point $B$ on the circle have coordinates $(x_2,y_2)$.

Step2: Calculate the slope $m$

The slope $m$ of the line segment $\overrightarrow{AB}$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$ (assuming $x_2
eq x_1$). If $x_2 = x_1$, the line is vertical and its equation is $x=x_1$.

Step3: Find the y - intercept $b$

We know that the line passes through the point $(x_1,y_1)$. Substitute $x = x_1$, $y=y_1$ and $m$ into $y=mx + b$. Then $y_1=m x_1 + b$, and we can solve for $b$: $b=y_1 - mx_1$.

Step4: Write the equation

The slope - intercept form of the line $\overrightarrow{AB}$ is $y=mx + b$, where $m=\frac{y_2 - y_1}{x_2 - x_1}$ and $b=y_1 - m x_1$.

Answer:

$y = mx + b$ where $m=\frac{y_2 - y_1}{x_2 - x_1}$ (if $x_2
eq x_1$) and $b=y_1 - mx_1$; if $x_2 = x_1$, the equation is $x=x_1$