### Video Transcript

Given that π΄π΅ is tangent to the circle and that π· and πΆ lie on the circle, determine the value of π₯ to the nearest tenth.

So letβs look at this diagram carefully. Weβre told that the line π΄π΅ is tangent to the circle. The line π΅π· intersects the circle in two places: the points πΆ and π·. And therefore, the line π΅π· is a secant. Weβve been given the length π΄π΅ explicitly β itβs 21 centimeters β and the length π΅π· in terms of π₯ and asked to determine its value.

In order to do so, we need to record the relationship that exists between the length of a tangent and the length of a secant that intersect at a point. The relationship is this: if a tangent and the secant intersect outside a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant and its external secant segment.

Now, thatβs quite a long statement. So letβs take it step by step in order to understand what it means in this question. Firstly, it says if a tangent and a secant intersect outside a circle, well, ours do. The line π΄π΅ and the line π΅π· intersect at the point π΅, which is outside the circle. Next, it talks about the square of the measure of the tangent. So in our question, that is π΄π΅ squared.

Next, weβre told that this is equal to the product of the measures of the secant and its external secant segment. The secant is the full line π΅π·. The external secant segment is just a portion of the secant outside the circle. So itβs the segment π΅πΆ.

So we have the equation π΄π΅ squared is equal to π΅π· multiplied by π΅πΆ. Letβs substitute in some values or expressions for each of these parts. π΄π΅ squared is 21 squared, π΅π· is the full length of that line: so itβs π₯ plus π₯ plus two, and then π΅πΆ is equal to π₯. Letβs simplify this equation. 21 squared is equal to 441. And on the right-hand side, the bracket simplifies to two π₯ plus two.

Next, letβs expand the bracket on the right-hand side. We now have 441 is equal to two π₯ squared plus two π₯. Finally, letβs collect all terms on the right-hand side of the equation. To do so, we need to subtract 441 from both sides. So we now have the equation two π₯ squared plus two π₯ minus 441 is equal to zero.

Now, this is a quadratic equation. And it certainly isnβt one that easily factorizes. So in order to solve it, we need to use the quadratic formula. The quadratic formula tells us that the solutions to the equation ππ₯ squared plus ππ₯ plus π equals zero are given by π₯ is equal to negative π plus or minus the square root of π squared minus four ππ all over two π.

For our equation, π and π are both equal to two and π is equal to negative 441. So substituting our values of π, π, and π into the quadratic formula gives π₯ is equal to negative two plus or minus the square root of two squared minus four multiplied by two multiplied by negative 441 all over two multiplied by two. Now, if you use your calculator in order to evaluate this, it will give two possibilities for the positive and the negative square root: either 14.3576 or negative 15.3576.

Remember that π₯ represents a length. Itβs the length of the segment π΅πΆ. And therefore, it must take a positive value. So the solution that we need is the positive one, 14.3576. Weβre also asked to give our answer to the nearest tenth. So we need to round it. The value of π₯ to the nearest tenth is 14.4.

Remember the key fact that we used in this question: if a tangent and a secant intersect outside a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant and its external secant segment.