### Video Transcript

In the figure, circles π½ and πΎ
are congruent. The arc π΄π΅ is congruent to the
arc πΆπ·. π΄π΅ is equal to three π₯ plus
seven centimetres and πΆπ· is equal to eight π₯ plus 12 centimetres. Find the length of π΄π΅.

Firstly, thereβs some notation in
the question that we need to be comfortable with: this notation here π΄π΅ and πΆπ·,
each with a circumflex over them. This notation refers to the minor
arc connecting the two points β the shortest distance around the circumference
between the two points. So those are the arcs that Iβve
highlighted in orange. The second use of π΄π΅ and πΆπ·
this time without a circumflex refers to the cord connecting these two points. So those are the straight line
segments between the two points β the ones that Iβve now marked in green.

Weβve been told in the question
that the two circles are congruent and also that the two minor arcs π΄π΅ and πΆπ·
are the same length. Weβre asked to determine the length
of the chord π΄π΅. In order to answer this question,
we need to recall a key fact about arcs and chords in congruent circles.

Here is the key fact. In congruent circles, two minor
arcs are congruent if and only if their corresponding chords are congruent. The term if and only if means that
this statement works both ways: two minor arcs are congruent if their corresponding
chords are congruent, but also two corresponding chords are congruent if the two
minor arcs are congruent.

So as we know that the two arcs
π΄π΅ and πΆπ· are congruent, the statement above tells us that this implies that the
chords π΄π΅ and πΆπ· are also congruent. Weβve been given expressions for
the length of the two chords in the question. So as theyβre congruent, we can
form an equation. Setting the two expressions equal
to one another gives the equation three π₯ plus seven is equal to eight π₯ plus
12.

Now, our overall objective in this
question is to find the length of the chord π΄π΅. And in order to do so, we need to
know the value of π₯. So we need to solve this equation
in order to determine π₯. As those terms involving π₯ on both
sides of the equation, the first step is to subtract three π₯ from both sides. This gives the equation seven is
equal to five π₯ plus 12. Next, Iβm going to subtract 12 from
each side of the equation. This gives negative five is equal
to five π₯. The final step to solve this
equation is to divide both sides by five. This tells me that π₯ is equal to
negative one.

Now remember the reason we needed
to know the value of π₯ was so that we could calculate the length of the chord
π΄π΅. Our expression for the length of
the chord π΄π΅ is three π₯ plus seven. Now, we know the value of π₯, we
can substitute this into our expression in order to calculate the length of the
chord π΄π΅.

We have then that the chord π΄π΅ is
equal to three multiplied by negative one plus seven. This is equal to negative three
plus seven, which is equal to four. So we have our answer to the
problem. The length of the chord π΄π΅ is
four centimetres.

Remember the key fact we used in
this question was that in congruent circles two minor arcs are congruent if and only
if their corresponding chords are also congruent.