### Video Transcript

Circle π has radius 11 centimeters. If πΆπ΄ is equal to 16.3 centimeters, what is π΄π΅? Answer to the nearest tenth.

So weβre given a diagram of a circle and then a triangle formed by connecting three points: π, which is the center of the circle; π΅, which is a point on the circumference; and π΄, which is an external point. Weβre asked to calculate the length of the line segment π΄π΅. Weβre given various pieces of information in the question. Firstly, that the length of the line πΆπ΄ is equal to 16.3 centimeters. Secondly, weβre told that the radius of the circle is equal to 11 centimeters. Remember, the radius of a circle is a line whose end points are the center of the circle and the point on the circumference. So in fact, there are two radii in the diagram, the lines ππ΅ and ππΆ.

Having added these labels to the diagram, we can now see that the length weβve been asked to calculate, π΄π΅, is in fact the third side of a triangle, triangle π΄ππ΅. And we know the length of the other two sides of the triangle. π΅π, remember, is a radius of the circle. So itβs 11 centimeters. And the full length of the side π΄π is 27.3 centimeters. Do we know anything else about this triangle? Well, in fact we do. π΄π΅ is a tangent to the circle. And a key fact about tangents to circles is this: a tangent to a circle is perpendicular to the radius at the point of contact. The radius at this point is the line ππ΅. And so we know that the lines ππ΅ and π΄π΅ are perpendicular. Therefore, this triangle is a right-angled triangle.

If we know the length of two sides of a right-angled triangle and we want to calculate the length of the third side, we can do this by applying the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In our triangle, this means that π΄π΅ squared plus 11 squared is equal to 27.3 squared. This gives an equation that we can solve in order to find the length of π΄π΅.

The first step is to evaluate both 11 squared and 27.3 squared. This gives π΄π΅ squared plus 121 is equal to 745.29. The next step is to subtract 121 from each side of the equation. We now have π΄π΅ squared is equal to 624.29. To find the value of π΄π΅, we need to take the square root of both sides. This gives π΄π΅ is equal to the square root of 624.29, which as a decimal is equal to 24.98579.

Remember, the question asked as to give our answer to the nearest tenth. So we need to round this decimal. Our answer to the problem is that the length of π΄π΅, to the nearest tenth, is 25.0 centimeters. Remember, the key fact we used in this question was that a tangent to a circle is perpendicular to the radius at the point of contact.