Question Video: Finding the Area of a Triangle Given a Circle Inscribed in a Quadrilateral | Nagwa Question Video: Finding the Area of a Triangle Given a Circle Inscribed in a Quadrilateral | Nagwa

# Question Video: Finding the Area of a Triangle Given a Circle Inscribed in a Quadrilateral Mathematics • Third Year of Preparatory School

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In the figure, a circle with radius 6 cm is inscribed in quadrilateral π΄π΅πΆπ·. If πΈ is the midpoint of line segment π΄π΅, π΄π΅ = 10 cm, and πΆπΊ = 6 cm, find the area of triangle π΅πΆπ.

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### Video Transcript

In the following figure, a circle with radius six centimeters is inscribed in quadrilateral π΄π΅πΆπ·. If πΈ is the midpoint of line segment π΄π΅, π΄π΅ equals 10 centimeters, and πΆπΊ equals six centimeters, find the area of triangle π΅πΆπ.

Letβs begin by adding any given lengths to our diagram. First, weβre 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 ππΈ is six centimeters, but we could choose any point on the circumference of the circle. So, we could write line segment ππ» is six centimeters. Similarly, ππΉ and ππΊ will all be six centimeters in length.

Next, weβre given three further pieces of information. Weβre told that the line segment π΄π΅ is 10 centimeters in length. Weβre also told that πΈ is the midpoint of line segment π΄π΅. So, π΄πΈ and πΈπ΅ must each be five centimeters in length. Then, weβre also told that the line segment between point πΆ and πΊ is six centimeters. Our job is to use this to find the area of triangle π΅πΆπ. And of course, thatβs 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 ππΉ is the radius of our circle. In fact, we can also see that line segment πΆπ΅, which meets the circle at point πΉ, is the tangent. So, line segment ππΉ must be perpendicular to line segment πΆπ΅. This means we know the perpendicular height of the triangle. If we call its base the line segment πΆπ΅, then the perpendicular height must be ππΉ, which is six centimeters.

So next, we need to find the length of line segment πΆπ΅. In order to find the length of line segment πΆπ΅, 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, thatβs the tangent segments πΈπ΅ and πΉπ΅ which meet at point π΅ β then they must be equal in length. So, the length of line segment πΈπ΅ must be equal to the length of line segment πΉπ΅. And remember, we said that since line segment π΄π΅ was 10 centimeters and point πΈ is exactly halfway along the line segment π΄π΅, then both π΄πΈ and πΈπ΅ are five centimeters. So, line segment πΉπ΅ must also be five centimeters.

And of course, if we look carefully, we see we can repeat this by looking at line segments πΉπΆ and πΆπΊ. πΉπΆ and πΆπΊ must be equal in length. So, line segment πΉπΆ is six centimeters. And so, weβre now able to find the length of line segment πΆπ΅. 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΅πΆπ 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΅πΆπ·, the area of π΅πΆπ is 33 square centimeters.

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