Video: Finding the Length of a Chord in a Circle by Solving Two Linear Equations

In the figure, circles 𝐽 and 𝐾 are congruent, 𝐴𝐡 β‰… 𝐢𝐷, 𝐴𝐡 = (3π‘₯ + 7) cm and 𝐢𝐷 = (8π‘₯ + 12) cm. Find the length of 𝐴𝐡.

03:56

Video Transcript

In the figure, circles 𝐽 and 𝐾 are congruent. The arc 𝐴𝐡 is congruent to the arc 𝐢𝐷. 𝐴𝐡 is equal to three π‘₯ plus seven centimetres and 𝐢𝐷 is equal to eight π‘₯ plus 12 centimetres. Find the length of 𝐴𝐡.

Firstly, there’s some notation in the question that we need to be comfortable with: this notation here 𝐴𝐡 and 𝐢𝐷, 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 I’ve highlighted in orange. The second use of 𝐴𝐡 and 𝐢𝐷 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 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 𝐴𝐡 and 𝐢𝐷 are the same length. We’re asked to determine the length of the chord 𝐴𝐡. 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 𝐴𝐡 and 𝐢𝐷 are congruent, the statement above tells us that this implies that the chords 𝐴𝐡 and 𝐢𝐷 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 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 𝐴𝐡. 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, I’m 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 𝐴𝐡. Our expression for the length of the chord 𝐴𝐡 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 𝐴𝐡.

We have then that the chord 𝐴𝐡 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 𝐴𝐡 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.

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