Question

refer to the figure below, which shows a circle with center c and a radius of r, and right triangle adc. find r if ab = 8 and ad = 12. answer exactly. r =

Explanation:

Step1: Apply tangent - secant relationship

Since $AD$ is tangent to the circle at $D$ and $AB$ is a secant - like segment, we can use the property of right - triangle and circle. In right - triangle $ADC$, $AC=AB + r=8 + r$, $AD = 12$, and $CD=r$.

Step2: Use Pythagorean theorem

By the Pythagorean theorem in right - triangle $ADC$, we have $AC^{2}=AD^{2}+CD^{2}$. Substitute $AC = 8 + r$, $AD = 12$, and $CD=r$ into the formula: $(8 + r)^{2}=12^{2}+r^{2}$.

Step3: Expand the left - hand side

Expand $(8 + r)^{2}$ using the formula $(a + b)^{2}=a^{2}+2ab + b^{2}$. So, $64+16r+r^{2}=144 + r^{2}$.

Step4: Solve for $r$

Subtract $r^{2}$ from both sides of the equation: $64+16r+r^{2}-r^{2}=144 + r^{2}-r^{2}$. This simplifies to $16r=144 - 64$. Then $16r = 80$. Divide both sides by 16: $r=\frac{80}{16}=5$.

Answer:

$5$