Question

1. solve for x.

(a) if x = 8 color the rest of the background blue.
(b) if x = 4 color the rest of the background purple.

Explanation:

Step1: Identify the triangle type

The triangle formed is a right - triangle (since the radius is perpendicular to the tangent at the point of contact). Let the radius be \( r = 4 \), the length of the tangent segment be \( 4\sqrt{3} \), and the hypotenuse of the right - triangle (the segment from the external point to the center) be \( x + 4 \).

Step2: Apply the Pythagorean theorem

According to the Pythagorean theorem, for a right - triangle with legs \( a = 4 \) and \( b = 4\sqrt{3} \), and hypotenuse \( c=x + 4 \), we have \( c^{2}=a^{2}+b^{2} \).
Substitute the values: \((x + 4)^{2}=4^{2}+(4\sqrt{3})^{2}\)
Calculate the right - hand side: \(4^{2}=16\) and \((4\sqrt{3})^{2}=4^{2}\times(\sqrt{3})^{2}=16\times3 = 48\). So \(4^{2}+(4\sqrt{3})^{2}=16 + 48=64\)
Then \((x + 4)^{2}=64\)
Take the square root of both sides: \(x + 4=\pm8\)
Since \(x\) represents a length, \(x+4>0\), so we consider \(x + 4 = 8\) or \(x+4=- 8\) (we discard \(x + 4=-8\) because \(x=-12\) does not make sense for a length).
If \(x + 4 = 8\), then \(x=8 - 4=4\)

Answer:

\( x = 4 \)