Geometry help!?

Question

Find the area of the segment of the circle shaded in blue. The radius of the circle is 20 units and the base of the triangle is 24 units. Use pi = 3.14 and round your answers to the nearest hundredth.
Show your work for credit.

http://i101.photobucket.com/albums/m53/yoobritt/06_06_18.gif

Answer 1

easy way to do this is find the area of the whole piece of the circle, then subtract the angle of the triangle.

Area of a piece of circle = (theta)r^2
change 110 degrees to 110 (pi)/ 180 = 11pi/18
Area = (11pi/18)(20)^2 = 400(11pi/18)

Area of triangle: note that the base is 24, and both other sides are radii or length 20. this tells you the triangle is isosceles and therefore the two unknown angles are equal
180 degrees - 110 degrees = 70 degrees
so, each smaller angle = 70/2 = 35 degrees.

You need that information because area of a triangle is (1/2)bh, but we dont yet know h.
let's split the triangle into two right triangles to find h.

sin (35degrees) = h/(radius = hypotenuse = 20)
h = 20 sin(35)

Triangle area then is (1/2)(24)(20sin(35)) = 240sin(35)

Area of shaded part of circle:
Area of whole circle piece - area of triangle
400(11pi/18) - 240sin(35 degrees)
pi = 3.14
so,
A(region) = (4400*3.14)/18 - 240sin(35degrees)
round to nearest hundredth.

Answer 2

Let the center of the circle be "O", and the radii forming the 110 degrees angle OA and OB.
Area of full circle= pi* r^2= 400 pi
Area of OAB= 400pi* 110/360
Since radii are sides of a triangle, the triangle is isoceles. Draw angle bisector to AB from AOB.
By properties of isoc triangle, this bisector also bisects AB and forms two congruent right triangles. From phythag thm, 12^2+ (bisector)^2=20^2 and bisector length=16. Then area of triangle AOB is 2*(1/2)*12*16= 192
Then blue segment = 400*pi*(110/360) - 192

Answer 3

the area segment of the circle shaded =

area of sector- area of triangle

area of sector = r^2/2 (pi/180)(central angle)

= pi r^2(110)/360

= 3.14(20)^2 (110)/360

= 383.8 square units

area of triangle = bh/2

b = 24

since perpendicular(h) from center to base of triangle bisects the base of triangle. From pythogoras theorem

h = r^2 - b^2/4

= 20^2 - 24^2/4

= 400 - 576/4

= 400 - 144

= 256

h = sqrt(256)

= 16

area of triangle = bh/2= 24*16/2

= 192 square units

area of shaded region = 383.8 - 192

= 191.8 square units

Answer 4

Area of sector = (110/360)(pi)(20)^2= 1100pi/9 units^2
area of triangle = 187.88
So area of blue segment is 1100pi/9 - 187.88 = 196.09 units^2.

I don't understand the number 24 in the blue area. It cannot be the length of the chord or the length of the arc.

The length of the chord is 2*20sin(55) = 32.76 units. Length of arc is r*theta = 20*110pi/180 = 38. 4 units.

Are you sure you copied this correctly?