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Given:• Points A, B, and C are points of tangency.• VZ = 24• AV = 14• The perimeter of AWV Z is 64.BzcWFind AW.281416

Given:• Points A, B, and C are points of tangency.• VZ = 24• AV = 14• The perimeter-example-1

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Since the segments taht are shown are all tangents to the circle, we can use the property that the tanget segments of the same circle and that have an external common point have the same length. From this, we see that the pairs AV and BV, BZ and CZ, CW and AW are tangets with common eterior points, thus:


\begin{gathered} AV=BV \\ BZ=CZ \\ CW=AW \end{gathered}

Than, we can see that that:


\begin{gathered} BV+BZ=VZ \\ BV+BZ=24 \\ AV=BV \\ AV+BZ=24 \\ 14+BZ=24 \\ BZ=10 \end{gathered}

Also:


\begin{gathered} CZ=BZ \\ CZ=10 \end{gathered}

So, we already know AV, BV, BZ and CZ. AW and CW are the same, and the sum of all of them is the perimeter, that we know is 64, so:


\begin{gathered} AV+BV+BZ+CZ+CW+AW=64 \\ 14+14+10+10+CW+CW=64 \\ 2CW=64-48 \\ 2CW=16 \\ CW=8 \\ AW=CW \\ AW=8 \end{gathered}

User Henry Pootle
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