Video Transcript
In the figure, circles 𝐽 and 𝐾
are congruent. The arc 𝐴𝐵 is congruent to the
arc 𝐶𝐷. 𝐴𝐵 is equal to three 𝑥 plus
seven centimetres and 𝐶𝐷 is equal to eight 𝑥 plus 12 centimetres. Find the length of 𝐴𝐵.
Firstly, there’s some notation in
the question that we need to be comfortable with: this notation here 𝐴𝐵 and 𝐶𝐷,
each with a circumflex over them. This notation refers to the minor
arc connecting the two points — the shortest distance around the circumference
between the two points. So those are the arcs that I’ve
highlighted in orange. The second use of 𝐴𝐵 and 𝐶𝐷
this time without a circumflex refers to the cord connecting these two points. So those are the straight line
segments between the two points — the ones that I’ve now marked in green.
We’ve been told in the question
that the two circles are congruent and also that the two minor arcs 𝐴𝐵 and 𝐶𝐷
are the same length. We’re asked to determine the length
of the chord 𝐴𝐵. 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
𝐴𝐵 and 𝐶𝐷 are congruent, the statement above tells us that this implies that the
chords 𝐴𝐵 and 𝐶𝐷 are also congruent. We’ve been given expressions for
the length of the two chords in the question. So as they’re congruent, we can
form an equation. Setting the two expressions equal
to one another gives the equation three 𝑥 plus seven is equal to eight 𝑥 plus
12.
Now, our overall objective in this
question is to find the length of the chord 𝐴𝐵. And in order to do so, we need to
know the value of 𝑥. So we need to solve this equation
in order to determine 𝑥. As those terms involving 𝑥 on both
sides of the equation, the first step is to subtract three 𝑥 from both sides. This gives the equation seven is
equal to five 𝑥 plus 12. Next, I’m going to subtract 12 from
each side of the equation. This gives negative five is equal
to five 𝑥. The final step to solve this
equation is to divide both sides by five. This tells me that 𝑥 is equal to
negative one.
Now remember the reason we needed
to know the value of 𝑥 was so that we could calculate the length of the chord
𝐴𝐵. Our expression for the length of
the chord 𝐴𝐵 is three 𝑥 plus seven. Now, we know the value of 𝑥, we
can substitute this into our expression in order to calculate the length of the
chord 𝐴𝐵.
We have then that the chord 𝐴𝐵 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 𝐴𝐵 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.
