if chord of 10 cm is drawn in a circle of 8cm, then find the area of both segments?

Question

• cant use trignumetry
• Difficult questions
• Based on areas of circles

Answer 1

draw a circle with a horizontal diameter which is 16 cm

draw the chord above the diameter which is 10 cm

draw two radii to each endpoint of the chord

you have an isosceles triangle formed by the two radii and the chord and you have a sector formed by the two radii connecting the endpoints of the chord along with the arc above the 10 cm chord

find the central angle of the chord using the law of cosines

10^2=8^2+8^2-2*8*8*cos(x) where x is the central angle

100=128-128cos(x)

-28=-128cos(x)

-28/-128=cos(x)

using arcos finder on your calculator, find x

-28/-128=0.21875

cos(x)=0.21875

using arcos finder...

x=77.36437491º

now find the area of the sector

A=(77.36437491/360)*∏*r^2

A=(0.2149)(3.14159)(64)

A=0.67512769*64

A=43.20817  sq cm

now find the area of the triangle

draw the vertical altitude which forms two right triangles

call the altitude "y" and use the Pythagorean Theorem

8^2=y^2+5^2 (the altitude bisects the 10 cm chord)

64-25=y^2

y^2=39

y=√39=6.24499

find the area of the triangle

A=(1/2)(10)(6.24499)

A=5*6.24499

A=31.22495 sq cm

subtract the area of the triangle from the area of the sector

43.20817-31.22495=11.98322 sq cm is the area above the 10 cm chord

find the area of the circle

A=∏*8^2

A=3.14159*64

A=201.06176 sq cm

subtract the area above the chord from the area of the circle to get the area below the chord

201.06176-11.98322=189.07854 sq cm for the area below the chord

Answer 2

Draw a circle with a chord going through it. See that the chord forms an Isosceles triangle with the center of the circle? Sides are radius, radius, chord length. It also creates a sector, or slice of the circle. Calculate the area of the circle, sector and triangle. Then add/subtract them to obtain the area of the segment and partial circle.