Question

determining the length of the radius
\\(\overline{lk}\\) is tangent to circle j at point k.
what is the length of the radius?
\\(\frac{167}{12}\\)
\\(\frac{85}{12}\\)
\\(\frac{6}{85}\\)

Explanation:

Step1: Recall Tangent - Radius Property

A tangent to a circle is perpendicular to the radius at the point of tangency. So, triangle \(IJK\) is a right triangle with \(\angle IKJ = 90^{\circ}\), where \(IJ=(r + 6)\), \(IK = 11\), and \(JK=r\).

Step2: Apply Pythagorean Theorem

In right triangle \(IJK\), by the Pythagorean theorem, \(IJ^{2}=IK^{2}+JK^{2}\). Substituting the values, we get \((r + 6)^{2}=11^{2}+r^{2}\).

Step3: Expand and Simplify the Equation

Expand \((r + 6)^{2}\): \(r^{2}+12r + 36=121+r^{2}\).
Subtract \(r^{2}\) from both sides: \(12r+36 = 121\).
Subtract 36 from both sides: \(12r=121 - 36=85\).
Solve for \(r\): \(r=\frac{85}{12}\).

Answer:

\(\frac{85}{12}\)