Video Transcript
Find length 𝐵𝐶 to the nearest integer.
Looking at the diagram we’ve been given, we can see that we have a circle. There is a tangent to the line 𝐴𝐷 and a secant segment 𝐴𝐶, which intersects the circle at point 𝐵 and point 𝐶. We’ve been given the length of the line segment from point 𝐴 to where the tangent meets the circle. And we want to determine the length of 𝐵𝐶, which is a segment of the secant. We also notice on the diagram that these lines indicate that the line segments 𝐴𝐵 and 𝐵𝐶 are of equal length. So whatever 𝐵𝐶 is, it’s the same as 𝐴𝐵. It’s also half the length of 𝐴𝐶.
As we’re working with the lengths of tangents and secants of a circle, we can recall the tangent–secant theorem. This is a special case of the intersecting secants theorem, which can be applied when one of the lines is a tangent. It’s as stated in the diagram. If there is a tangent 𝐸𝐶 to a circle and a secant 𝐸𝐴 to that circle, which intersects the circle first at 𝐵 and then at 𝐴, then 𝐸𝐶 squared is equal to 𝐸𝐵 multiplied by 𝐸𝐴.
Let’s see if we can identify the various lengths for our diagram. 𝐸𝐶 is the length of the tangent segment from the point outside the circle to where the tangent meets the circle. So, in our diagram, that’s the length 𝐴𝐷. 𝐸𝐵 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, that’s the line segment 𝐴𝐵. And then 𝐸𝐴 is the secant segment from the point outside the circle to the second point where the secant meets the circle. So, in our diagram, that’s 𝐴𝐶. We therefore have the equation 𝐴𝐷 squared equals 𝐴𝐵 multiplied by 𝐴𝐶.
Now, we know the length of 𝐴𝐷. It’s 164 centimeters. We also know that 𝐴𝐵 is the same length as 𝐵𝐶, which is the length we’re asked to find. We also stated earlier that 𝐵𝐶 is half the length of 𝐴𝐶. And so it follows that two 𝐵𝐶 is equal to 𝐴𝐶. We therefore have the equation 164 squared equals 𝐵𝐶 multiplied by two 𝐵𝐶. 164 squared is 26,896. And on the right-hand side, 𝐵𝐶 multiplied by two 𝐵𝐶 is two 𝐵𝐶 squared. Dividing both sides of this equation by two, we have that 𝐵𝐶 squared is equal to 13,448. We can then find the value of 𝐵𝐶 by taking the square root of each side of the equation, taking only the positive value as 𝐵𝐶 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 𝐵𝐶 is 116 centimeters. So, by using the tangent–secant theorem, we were able to show that the length of 𝐵𝐶, which is a segment of the secant 𝐴𝐶 to this circle, to the nearest integer is 116 centimeters.
