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: Recall tangent - radius property

A line tangent to a circle is perpendicular to the radius at the point of tangency. So, triangle $JKL$ is a right - triangle with $\angle{JKL}=90^{\circ}$.

Step2: Apply the Pythagorean theorem

In right - triangle $JKL$, by the Pythagorean theorem $JL^{2}=JK^{2}+KL^{2}$. We know that $KL = 24$, $JL=r + 16$, and $JK=r$. So, $(r + 16)^{2}=r^{2}+24^{2}$.

Step3: Expand the left - hand side

Expand $(r + 16)^{2}$ using the formula $(a + b)^{2}=a^{2}+2ab + b^{2}$. We get $r^{2}+32r+256=r^{2}+576$.

Step4: Simplify the equation

Subtract $r^{2}$ from both sides of the equation: $32r+256 = 576$.

Step5: Solve for $r$

Subtract 256 from both sides: $32r=576 - 256=320$. Then divide both sides by 32: $r=\frac{320}{32}=10$.

Answer:

10 units