line segment kl is tangent to circle j at point k. what is the length of the radius, r? 8 units 10 units 12…
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
Answer
Explanation:
Step1: Recall tangent - radius property
A tangent to a circle is perpendicular to the radius at the point of tangency. So, triangle JKL is a right - triangle with right - angle at K.
Step2: Apply the Pythagorean theorem
In right - triangle JKL, by the Pythagorean theorem (a^{2}+b^{2}=c^{2}), where (JK = r), (KL = 24), and (JL=r + 16). So, (r^{2}+24^{2}=(r + 16)^{2}).
Step3: Expand the right - hand side
Expand ((r + 16)^{2}) using the formula ((a + b)^{2}=a^{2}+2ab + b^{2}). We get (r^{2}+24^{2}=r^{2}+32r+256).
Step4: Simplify the equation
Subtract (r^{2}) from both sides of the equation: (r^{2}-r^{2}+576=r^{2}-r^{2}+32r + 256). This simplifies to (576=32r+256).
Step5: Solve for r
Subtract 256 from both sides: (576−256=32r), so (320 = 32r). Then divide both sides by 32: (r=\frac{320}{32}=10).
Answer:
10 units