find the value of x. assume that segments that appear to be tangent are tangent. round your answer to the…

Explanation:

Step1: Apply tangent - radius property

Since segments that appear to be tangent are tangent, a tangent to a circle is perpendicular to the radius at the point of tangency. So, triangle $MLN$ is a right - triangle with hypotenuse $ML$ and legs $LN$ and the radius $MN$.

Step2: Use the Pythagorean theorem

In right - triangle $MLN$, by the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = ML=32 + 13=45$ (the distance from the external point $L$ to the center of the circle $M$ is the sum of the length of the non - radius part and the radius) and $a = MN = 13$, and $b=x$. So, $x=\sqrt{ML^{2}-MN^{2}}$.

Step3: Substitute values and calculate

Substitute $ML = 45$ and $MN = 13$ into the formula: $x=\sqrt{45^{2}-13^{2}}=\sqrt{(45 + 13)(45 - 13)}=\sqrt{58\times32}=\sqrt{1856}\approx43.08$.

Answer:

$43.08$