Question

in the diagram, \overleftrightarrow{ac} and \overleftrightarrow{bf} are tangent to circle d at points a and e, respectively. \overleftrightarrow{ac} and \overleftrightarrow{bf}are tangent to circle g at points c and f, respectively. which of the following statements is false?

answer attempt 1 out of 2
\bigcirc \\ ba = be \qquad \bigcirc \\ cb = bf
\bigcirc \\ ab = ef \qquad \bigcirc \\ gc = gf

Explanation:

Step1: Recall tangent segment theorem

For a circle, two tangent segments from a single external point to the circle are congruent.

Step2: Analyze option $BA=BE$

Point $B$ is external to circle $D$; $\overrightarrow{AC}$ (tangent at $A$) and $\overrightarrow{BF}$ (tangent at $E$) are tangents to circle $D$ from $B$. By tangent segment theorem: $BA=BE$. This statement is true.

Step3: Analyze option $CB=BF$

Point $B$ is external to circle $G$; $\overrightarrow{AC}$ (tangent at $C$) and $\overrightarrow{BF}$ (tangent at $F$) are tangents to circle $G$ from $B$. By tangent segment theorem: $CB=BF$. This statement is true.

Step4: Analyze option $AB=EF$

$BA=BE$ (from Step2) and $CB=BF$ (from Step3), but $EF = BF - BE = CB - AB$. $EF$ equals $CB-AB$, not $AB$, unless $CB=2AB$, which is not guaranteed by the diagram or given information. This statement is false.

Step5: Analyze option $GC=GF$

$GC$ and $GF$ are both radii of circle $G$. All radii of a circle are congruent, so $GC=GF$. This statement is true.

Answer:

$AB = EF$