### Video Transcript

In the following figure, a circle with radius six centimeters is inscribed in quadrilateral π΄π΅πΆπ·. If πΈ is the midpoint of line segment π΄π΅, π΄π΅ equals 10 centimeters, and πΆπΊ equals six centimeters, find the area of triangle π΅πΆπ.

Letβs begin by adding any given lengths to our diagram. First, weβre told that the radius of the circle is six centimeters. In other words, the length of any line segment that joins the center π to a point on its circumference is six centimeters in length. For instance, line segment ππΈ is six centimeters, but we could choose any point on the circumference of the circle. So, we could write line segment ππ» is six centimeters. Similarly, ππΉ and ππΊ will all be six centimeters in length.

Next, weβre given three further pieces of information. Weβre told that the line segment π΄π΅ is 10 centimeters in length. Weβre also told that πΈ is the midpoint of line segment π΄π΅. So, π΄πΈ and πΈπ΅ must each be five centimeters in length. Then, weβre also told that the line segment between point πΆ and πΊ is six centimeters. Our job is to use this to find the area of triangle π΅πΆπ. And of course, thatβs this triangle here.

So, we begin by recalling that the area of a triangle is half the length of its base multiplied by its height. Remember, its base and its height must be perpendicular lengths. Now, this is really useful because we do know a little bit more about some of the line segments within this circle. We know that the tangent to a circle will meet the radius at 90 degrees. And we mentioned earlier that line segment ππΉ is the radius of our circle. In fact, we can also see that line segment πΆπ΅, which meets the circle at point πΉ, is the tangent. So, line segment ππΉ must be perpendicular to line segment πΆπ΅. This means we know the perpendicular height of the triangle. If we call its base the line segment πΆπ΅, then the perpendicular height must be ππΉ, which is six centimeters.

So next, we need to find the length of line segment πΆπ΅. In order to find the length of line segment πΆπ΅, we need to quote another fact about tangents. In particular, if we have a pair of tangent segments that meet at a point outside the circle β in here, thatβs the tangent segments πΈπ΅ and πΉπ΅ which meet at point π΅ β then they must be equal in length. So, the length of line segment πΈπ΅ must be equal to the length of line segment πΉπ΅. And remember, we said that since line segment π΄π΅ was 10 centimeters and point πΈ is exactly halfway along the line segment π΄π΅, then both π΄πΈ and πΈπ΅ are five centimeters. So, line segment πΉπ΅ must also be five centimeters.

And of course, if we look carefully, we see we can repeat this by looking at line segments πΉπΆ and πΆπΊ. πΉπΆ and πΆπΊ must be equal in length. So, line segment πΉπΆ is six centimeters. And so, weβre now able to find the length of line segment πΆπ΅. It will be the sum of these two measurements. It will be six centimeters plus five centimeters, which is, of course, equal to 11 centimeters. We now have the length of the base of our triangle and its perpendicular height. Its base is 11 centimeters, whilst its height is six centimeters.

So, the area of triangle π΅πΆπ is a half times 11 times six. One-half times six is, of course, equal to three. And so, the area is 11 times three, which is 33 or 33 square centimeters. Given the information about our circle and the quadrilateral π΄π΅πΆπ·, the area of π΅πΆπ is 33 square centimeters.