Question

true or false? the segment \\(\overline{ab}\\) is congruent to the segment \\(\overline{bc}\\). \\(\bigcirc\\) a. true \\(\bigcirc\\) b. false

Explanation:

Step1: Identify circle properties

$\overline{AC}$ is a diameter of the circle with center $O$, so $OA = OC = r$ (radius). $\overline{OB}$ is a radius perpendicular to $\overline{AC}$ at point $B$.

Step2: Calculate segment lengths

Let $OA = OC = r$. Then $AB = OA + OB$? No, correct: $AB = OA + OB$ is wrong. $AB = OA + OB$ is incorrect; actually, $AB = OA + OB$ is wrong. Wait, $O$ is on $\overline{AC}$, so $AC = 2r$, $OB$ is perpendicular to $AC$ at $B$. $OB = r$, so using Pythagoras in $\triangle OBC$: $BC = \sqrt{OC^2 - OB^2} = \sqrt{r^2 - r^2}$? No, wait no: $O$ is the center, $\overline{AC}$ is diameter, so $O$ is the midpoint of $\overline{AC}$, so $AO = OC = r$. $\overline{OB}$ is a radius, so $OB = r$, and $\overline{OB} \perp \overline{AC}$ at $B$. Then $AB = AO + OB$ is wrong, $O$ is between $A$ and $C$, so $AB = AO + OB$ is incorrect. Correct: $A---O---C$, $B$ is a point on $AC$ where $OB \perp AC$. So $AO = r$, $OB = r$, so $AB = AO + OB$ is wrong, $O$ is between $A$ and $B$? No, the diagram shows $O$ is on $AC$, $B$ is between $O$ and $C$? Wait no, the diagram: $A$ is on left, $C$ on right, $O$ is center (between $A$ and $C$), $B$ is a point on $AC$ closer to $O$, with $OB$ perpendicular to $AC$, and the other end of $OB$ is on the circle. So $AB = AO + OB$ no, $AO = r$, $OB = r$, $BC = OC - OB$? No, $OC = r$, $OB$ is perpendicular, so $AB = AO + OB$ is wrong. Let's assign values: let $r = 1$, so $AO = OC = 1$. $OB = 1$ (radius). Then in $\triangle OBC$, right-angled at $B$, $OC = 1$, $OB = 1$, so $BC = \sqrt{OC^2 - OB^2} = \sqrt{1 - 1} = 0$? No, that can't be. Wait, no, the other end of $OB$ is a point on the circle, so $OB$ is a radius, so the length from $O$ to that point is $r$, so $OB$ is the segment from $O$ to $B$, so $OB$ is not the radius, the segment from $O$ to the other point is the radius. Oh right! $B$ is the foot of the perpendicular from the circle point to $AC$, so $OB$ is not a radius, the segment from $O$ to the top point is the radius. Let $O$ be $(0,0)$, $A(-1,0)$, $C(1,0)$ (radius 1). The top point is $(0,1)$, so $B$ is $(0,0)$? No, no, the diagram shows $B$ is on $AC$, not at $O$. Wait, the top point is $(x,y)$, on the circle, so $x^2 + y^2 = 1$, and $OB$ is perpendicular to $AC$, $AC$ is horizontal, so $OB$ is vertical, so $B$ has the same x-coordinate as $O$, so $B$ is $(0, b)$, on $AC$ (the x-axis), so $B$ is $(0,0)$? No, the diagram shows $B$ is between $O$ and $C$, so $B$ is $(b, 0)$ where $0 < b < 1$, and the top point is $(b, k)$, which is on the circle, so $b^2 + k^2 = 1$, and $OB$ is perpendicular to $AC$, $AC$ is horizontal, so $OB$ is vertical, which matches. Then $AB =$ distance from $A(-1,0)$ to $B(b,0) = 1 + b$, $BC =$ distance from $B(b,0)$ to $C(1,0) = 1 - b$. Since $b > 0$, $1 + b
eq 1 - b$, so $AB
eq BC$.

Step3: Compare segment lengths

Since $AB = 1 + b$ and $BC = 1 - b$ with $b > 0$, $AB > BC$, so they are not congruent.

Answer:

B. False