Video Transcript
A circle has a radius of 10
centimeters. A chord of length 14 centimeters is
drawn. Find the area of the major segment,
giving the answer to the nearest square centimeter.
We are told that the circle has a
radius of 10 centimeters. A chord of length 14 centimeters is
drawn on the circle. If we let the two ends of the chord
be points 𝐴 and 𝐵 and the center point 𝑂, then the area of the minor segment is
equal to the area of the sector minus the area of the triangle. In order to calculate both of
these, we firstly need to work out the central angle 𝜃. This can be done in either radians
or degrees. In this question, we will use
radians. So, it is important that our
calculator is in the correct mode. The area of a sector, when 𝜃 is in
radians, is equal to a half 𝑟 squared 𝜃. And the area of a triangle is equal
to a half 𝑟 squared sin 𝜃. This can be simplified by
factoring, giving us the area of the segment equal to a half 𝑟 squared multiplied
by 𝜃 minus sin 𝜃.
We can now calculate the angle 𝜃
by using right-angle trigonometry or the cosine rule. In order to calculate the angle in
any triangle using the cosine rule, we use the following formula. Cos of 𝐴 is equal to 𝑏 squared
plus 𝑐 squared minus 𝑎 squared divided by two 𝑏𝑐, where 𝑎, 𝑏, and 𝑐 are the
three lengths of the triangle and 𝐴 is the one opposite the angle we’re trying to
work out. Substituting in our values gives us
cos of 𝜃 equals 10 squared plus 10 squared minus 14 squared over two multiplied by
10 multiplied by 10. This simplifies to cos of 𝜃 equals
one fiftieth. Ensuring that our calculator is in
radian mode, 𝜃 is equal to the inverse cos of one fiftieth. This is equal to 1.55079 and so on
radians.
We can now substitute this value
into our formula for the area of a segment. The area of the minor segment is
equal to 27.5497 and so on. We have been asked to calculate the
area of the major segment. This is the area of the whole
circle minus the area of the minor segment. The area of a circle is equal to
𝜋𝑟 squared. As our radius is equal to 10
centimeters, the area is equal to 100𝜋. We need to subtract 27.5497 and so
on from this. This is equal to 286.6095 and so
on. We’re asked to round our answer to
the nearest square centimeter. The deciding number is the six in
the tenths column. So, we round up to 287 square
centimeters. This is the area of the major
segment in the circle.
