Question

1. what is the length of the tangent drawn from p(6, 8) to a circle centered at c (3, 4) with radius 2?

Explanation:

Step1: Calculate the distance between point P and the center of the circle.

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $(x_1,y_1)=(3,4)$ and $(x_2,y_2)=(6,8)$. So $d=\sqrt{(6 - 3)^2+(8 - 4)^2}=\sqrt{3^2 + 4^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step2: Use the Pythagorean - theorem to find the length of the tangent.

Let the radius of the circle be $r = 2$, the distance from the external point to the center of the circle be $d = 5$, and the length of the tangent be $l$. According to the Pythagorean - theorem $l=\sqrt{d^{2}-r^{2}}$. Substitute $d = 5$ and $r = 2$ into the formula, we get $l=\sqrt{5^{2}-2^{2}}=\sqrt{25 - 4}=\sqrt{21}$.

Answer:

$\sqrt{21}$