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, and complete each step below. if you need help, follow these instructions 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 $(x_2,y_2)$. For simplicity, assume $A=(0,0)$ and radius $r = 5$, and $B=(3,4)$ (since $3^{2}+4^{2}=5^{2}$).

Step2: Calculate the slope $m$

The slope $m$ of the line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $x_1 = 0,y_1=0,x_2 = 3,y_2 = 4$, so $m=\frac{4 - 0}{3 - 0}=\frac{4}{3}$.

Step3: Find the $y -$intercept $b$

Since the line passes through the origin $(0,0)$, substituting $x = 0,y = 0$ and $m=\frac{4}{3}$ into $y=mx + b$, we get $0=\frac{4}{3}\times0 + b$, so $b = 0$.

Step4: Write the equation

The slope - intercept form of the line $\overrightarrow{AB}$ is $y=\frac{4}{3}x+0$, or simply $y=\frac{4}{3}x$.

Answer:

$y=\frac{4}{3}x$