### Video Transcript

In the following figure, find the value of π₯.

Letβs look at the diagram carefully. It consists of a circle with two chords: the lines π΄π΅ and πΆπ·. The two chords intersect at a point πΈ in the interior of the circle. Weβre asked to find the value of π₯, 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 πΆπ· β so πΆπΈ and πΈπ· β is equal to the product of the two segments of the chord π΄π΅ β so π΄πΈ and πΈπ΅. We have πΆπΈ multiplied by πΈπ· is equal to π΄πΈ multiplied by πΈπ΅.

As weβve been given expressions for the length of each of these chord segments, we can now substitute them to give an equation in terms of π₯. This gives π₯ multiplied by π₯ plus 12 for the product of the chord segments for the green chord πΆπ· is equal to π₯ plus eight multiplied by π₯ plus three for the product of the segments of the pink chord π΄π΅.

Now, letβs simplify this equation by expanding the brackets. On the left-hand side, we have π₯ squared plus 12π₯ and on the right-hand side π₯ squared plus eight π₯ plus three π₯ plus 24. And you can do this expansion however youβre 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 π₯. And itβs currently a quadratic equation. However, we have just a single π₯ squared on each side of the equation. And therefore, these cancel each other out directly. So now, this has become a linear equation.

Letβs simplify the right-hand side by grouping together the π₯ terms. We now have 12π₯ is equal to 11π₯. Thatβs eight π₯ plus three π₯ plus 24. To solve this equation, we need to subtract 11π₯ from both sides. Subtracting 11π₯ from 12π₯ just gives one π₯ or π₯. And on the right-hand side, weβre just left with 24. So in fact, weβve solved the equation to give π₯ 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.