In the following figure, find the value of 𝑥.
Let’s look at the diagram carefully. It consists of a circle with two chords: the lines 𝐴𝐵 and 𝐶𝐷. The two chords intersect at a point 𝐸 in the interior of the circle. We’re asked to find the value of 𝑥, which has been used in expressions for the lengths of the segments of the two chords.
In order to answer this question, we need to recall a key fact about the lengths of segments of intersecting chords. The fact we need is this: if two chords intersect in a circle, then the products of the lengths of the chord segments are equal.
In our circle, this means that the product of the two segments of the chords 𝐶𝐷 — so 𝐶𝐸 and 𝐸𝐷 — is equal to the product of the two segments of the chord 𝐴𝐵 — so 𝐴𝐸 and 𝐸𝐵. We have 𝐶𝐸 multiplied by 𝐸𝐷 is equal to 𝐴𝐸 multiplied by 𝐸𝐵.
As we’ve been given expressions for the length of each of these chord segments, we can now substitute them to give an equation in terms of 𝑥. This gives 𝑥 multiplied by 𝑥 plus 12 for the product of the chord segments for the green chord 𝐶𝐷 is equal to 𝑥 plus eight multiplied by 𝑥 plus three for the product of the segments of the pink chord 𝐴𝐵.
Now, let’s simplify this equation by expanding the brackets. On the left-hand side, we have 𝑥 squared plus 12𝑥 and on the right-hand side 𝑥 squared plus eight 𝑥 plus three 𝑥 plus 24. And you can do this expansion however you’re most comfortable expanding brackets, for example, using the FOIL method.
Now, I would like to solve this equation in order to find the value of 𝑥. And it’s currently a quadratic equation. However, we have just a single 𝑥 squared on each side of the equation. And therefore, these cancel each other out directly. So now, this has become a linear equation.
Let’s simplify the right-hand side by grouping together the 𝑥 terms. We now have 12𝑥 is equal to 11𝑥. That’s eight 𝑥 plus three 𝑥 plus 24. To solve this equation, we need to subtract 11𝑥 from both sides. Subtracting 11𝑥 from 12𝑥 just gives one 𝑥 or 𝑥. And on the right-hand side, we’re just left with 24. So in fact, we’ve solved the equation to give 𝑥 is equal to 24. Therefore, our answer to the problem is 24.
Remember the key fact that we used in this question was if two chords intersect in a circle, then the products of the lengths of the chord segments are equal.