### Video Transcript

Find length ๐ต๐ถ to the nearest integer.

Looking at the diagram weโve been given, we can see that we have a circle. There is a tangent to the line ๐ด๐ท and a secant segment ๐ด๐ถ, which intersects the circle at point ๐ต and point ๐ถ. Weโve been given the length of the line segment from point ๐ด to where the tangent meets the circle. And we want to determine the length of ๐ต๐ถ, which is a segment of the secant. We also notice on the diagram that these lines indicate that the line segments ๐ด๐ต and ๐ต๐ถ are of equal length. So whatever ๐ต๐ถ is, itโs the same as ๐ด๐ต. Itโs also half the length of ๐ด๐ถ.

As weโre working with the lengths of tangents and secants of a circle, we can recall the tangentโsecant theorem. This is a special case of the intersecting secants theorem, which can be applied when one of the lines is a tangent. Itโs as stated in the diagram. If there is a tangent ๐ธ๐ถ to a circle and a secant ๐ธ๐ด to that circle, which intersects the circle first at ๐ต and then at ๐ด, then ๐ธ๐ถ squared is equal to ๐ธ๐ต multiplied by ๐ธ๐ด.

Letโs see if we can identify the various lengths for our diagram. ๐ธ๐ถ is the length of the tangent segment from the point outside the circle to where the tangent meets the circle. So, in our diagram, thatโs the length ๐ด๐ท. ๐ธ๐ต is the length of the secant segment from the point outside the circle to where the secant first meets the circle. So, on our diagram, thatโs the line segment ๐ด๐ต. And then ๐ธ๐ด is the secant segment from the point outside the circle to the second point where the secant meets the circle. So, in our diagram, thatโs ๐ด๐ถ. We therefore have the equation ๐ด๐ท squared equals ๐ด๐ต multiplied by ๐ด๐ถ.

Now, we know the length of ๐ด๐ท. Itโs 164 centimeters. We also know that ๐ด๐ต is the same length as ๐ต๐ถ, which is the length weโre asked to find. We also stated earlier that ๐ต๐ถ is half the length of ๐ด๐ถ. And so it follows that two ๐ต๐ถ is equal to ๐ด๐ถ. We therefore have the equation 164 squared equals ๐ต๐ถ multiplied by two ๐ต๐ถ. 164 squared is 26,896. And on the right-hand side, ๐ต๐ถ multiplied by two ๐ต๐ถ is two ๐ต๐ถ squared. Dividing both sides of this equation by two, we have that ๐ต๐ถ squared is equal to 13,448. We can then find the value of ๐ต๐ถ by taking the square root of each side of the equation, taking only the positive value as ๐ต๐ถ is a length. Evaluating this on a calculator gives 115.9655 continuing.

The question specifies that we should give our answer to the nearest integer. So, rounding this value, we have that the length of ๐ต๐ถ is 116 centimeters. So, by using the tangentโsecant theorem, we were able to show that the length of ๐ต๐ถ, which is a segment of the secant ๐ด๐ถ to this circle, to the nearest integer is 116 centimeters.