Question

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as shown in the figure, $overline{ab}$ is a chord of $odot o$. $overline{od}perpoverline{ab}$ at point $c$, intersecting $odot o$ at point $d$. draw $overline{oa}$. if $ab = 8$, and $cd = 2$, then the radius of $odot o$ is.

Explanation:

Step1: Use the property of perpendicular - chord bisector

Since $OD\perp AB$ and $OD$ is a radius of the circle $\odot O$, by the perpendicular - chord bisector theorem, $AC = \frac{1}{2}AB$. Given $AB = 8$, then $AC=\frac{8}{2}=4$.

Step2: Let the radius of the circle be $r$

Let the radius of $\odot O$ be $r$, then $OA=r$ and $OC=r - CD$. Given $CD = 2$, so $OC=r - 2$.

Step3: Apply the Pythagorean theorem in right - triangle $OAC$

In right - triangle $OAC$, according to the Pythagorean theorem $OA^{2}=AC^{2}+OC^{2}$. Substitute $OA = r$, $AC = 4$, and $OC=r - 2$ into the formula: $r^{2}=4^{2}+(r - 2)^{2}$.

Step4: Expand and solve the equation

Expand $(r - 2)^{2}$: $r^{2}=16+r^{2}-4r + 4$.
Subtract $r^{2}$ from both sides of the equation: $0=16-4r + 4$.
Combine like terms: $4r=20$.
Solve for $r$: $r = 5$.

Answer:

5