Question

line segment kl is tangent to circle j at point k. what is the length of the radius, r? 8 units 10 units 12 units 16 units

Explanation:

Step1: Apply the tangent - radius property

Since \(KL\) is tangent to the circle \(J\) at \(K\), \(JK\perp KL\). So, \(\triangle JKL\) is a right - triangle with \(JL=(r + 16)\), \(JK = r\), and \(KL = 24\).

Step2: Use the Pythagorean theorem

By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (in \(\triangle JKL\), \(a = r\), \(b = 24\), \(c=r + 16\)), we have \(r^{2}+24^{2}=(r + 16)^{2}\).
Expand \((r + 16)^{2}\) using the formula \((A + B)^{2}=A^{2}+2AB + B^{2}\), so \(r^{2}+24^{2}=r^{2}+32r+256\).
Subtract \(r^{2}\) from both sides: \(576=32r + 256\).
Subtract 256 from both sides: \(32r=576 - 256=320\).
Divide both sides by 32: \(r = 10\).

Answer:

10 units