Given that 𝐴𝐵 equals 42 centimetres and 𝐸𝐶 equals 10 centimetres in the two concentric circles shown below, find the length of line segment 𝐴𝐶.
So let’s have a look at the diagram of the circles and the lines and see if we can figure out anything about their properties. We’re told that the circles are concentric, which means that they share the same centre, in this case, the point 𝑂. We can also see that we have some chords in our circles, the chord 𝐷𝐶 in the smaller circle and the chord 𝐴𝐵 in the larger circle. A chord is a straight line segment whose endpoints lie on the circle.
Let’s also recall another key fact about chords. And that is that the part perpendicular bisector of a chord passes through the centre of the circle. And a reminder that perpendicular means at 90 degrees. And to bisect means to cut exactly in half. If we look at our diagram, we can see that the line 𝑂𝐸 cuts the chord 𝐷𝐶 and the chord 𝐴𝐵 at 90 degrees. It will also bisect these lines. That means that the line 𝐷𝐸 is equal to the line 𝐸𝐶. And the line 𝐵𝐸 is equal to the line 𝐴𝐸. So if we fill in the fact that we’re given, that 𝐸𝐶 is 10 centimetres, then the line 𝐸𝐷 must also be 10 centimetres. Since we’ve established that these lines are the same length.
To find the length of 𝐴𝐶 then, we need to use a fact we’re given that 𝐴𝐵 is 42 centimetres. So if we call our unknown length 𝑥, then we notice that we still have an unknown length on this line segment 𝐴𝐵. However, recalling that our line 𝑂𝐸 is a perpendicular bisector, this means that our length 𝐵𝐸 is exactly the same length as the line 𝐴𝐸. Which means that the length of 𝐵𝐷 can also be defined as 𝑥.
Therefore, we can write that 𝑥 plus 10 plus 10 plus 𝑥 equals 42. This simplifies to two 𝑥 plus 20 equals 42. Subtracting 20 from both sides of our equation will give us two 𝑥 equals 22. And 𝑥 is equal to 11. Therefore, our final answer is that our line 𝐴𝐶 is 11 centimetres.