In a triangle ^@ABC^@, ^@AD^@ is a median. ^@F^@ is a point on ^ | Math Question and Answer

Answer:

9 cm

Step by Step Explanation:

We are given that $A D$ $A D$ is the median of $△ A B C$ $△ A B C$ and $E$ $E$ is the midpoint of $A D .$ $A D .$

Let us draw a line $D G$ $D G$ parallel to $B F$ $B F$ .
Now, in $\triangle ADG$ , $E$ is the midpoint of $AD$ and $EF \parallel DG.$

By converse of the midpoint theorem we have $F$ as midpoint of $AG.$ $\begin{aligned} \implies & AF = FG && \ldots \text(1) \end{aligned}$

Similarly, in $\triangle BCF$ , $D$ is the midpoint of $BC$ and $DG \parallel BF.$

By converse of midpoint theorem we have $G$ is midpoint of $CF.$ $\begin{aligned} \implies FG = GC && \ldots \text(2) \end{aligned}$
From equations (1) and (2), we get $\begin{aligned} AF = FG = GC && \ldots \text(3) \end{aligned}$ Also, from the figure we see that $\begin{aligned} AF + FG + GC = AC \\ AF + AF + AF = AC && \text { [from (3)] } \\ 3 AF = AC \end{aligned}$
We are given that $AF$ = 3 cm.
Thus, $AC = 3 AF = 3 \times 3 \text{ cm} = 9 \text{ cm}$ .

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