Exercise 42 Page 665

A part of a circle bounded by an arc and the segment joining its endpoints is a segment of a circle.

To find the area of a segment for a minor arc, we need to draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed.

For the given diagram, we will find the area of the sector and then the area of the triangle. Finally, we will find their difference to find the area of the segment.

Area of the Sector

The area of a sector of a circle is the product of the measure of the arc divided by 360 and the area of the circle.

With this in mind, let's consider the given diagram.

We can see that the radius of the circle is 9ft. Also, we can see that the radii form an isosceles triangle and that one of the angles opposite the legs has a measure of 45. Therefore, we can find the measure of the central angle that corresponds to our arc. 180-2* 45= 90 Recall that the measure of an arc is the same as the measure of its corresponding central angle. Therefore, the measure of our arc is also 90. Let's consider how these pieces of information fit in the diagram.

We have all the information we need to calculate the area of the sector. Let's do it! The area of the sector is 20.25π ft^2.

Area of the Triangle

The area of a triangle can be found by taking half the product of two side lengths and the sine of the included angle. With this in mind, let's consider the triangle in our diagram.

The length of both of the known sides is 9ft and the included angle measures 90^(∘). This is enough information to find the area of the triangle. The area of the triangle is 40.5ft^2.

Area of the Segment

Finally, to obtain the area of the segment we need to subtract the area of the triangle from the area of the sector. The area of the segment, to the nearest tenth, is 23.1ft^2.