Answer:
To find the area of the segment, we need to subtract the area of the triangle formed by the radii and the chord from the area of the sector formed by the central angle.
The radius of the circle is given as 3 cm, so the length of the chord is also 3 cm as it bisects the diameter. We can use the Pythagorean theorem to find the length of the half chord:
Half chord
=
(
radius
)
2
−
(
distance from center to chord
)
2
Half chord=
(radius)
2
−(distance from center to chord)
2
The distance from the center to the chord is half the length of the chord, which is 1.5 cm. So,
Half chord
=
(
3
cm
)
2
−
(
1.5
cm
)
2
=
6.75
cm
Half chord=
(3 cm)
2
−(1.5 cm)
2
=
6.75
cm
The area of the sector formed by the central angle of 90 degrees is:
Area of sector
=
central angle
36
0
∘
×
�
(
radius
)
2
=
9
0
∘
36
0
∘
×
�
(
3
cm
)
2
=
9
4
�
cm
2
Area of sector=
360
∘
central angle
×π(radius)
2
=
360
∘
90
∘
×π(3 cm)
2
=
4
9
π cm
2
The area of the triangle formed by the radii and the chord is:
Area of triangle
=
1
2
(
half chord
)
(
radius
)
=
1
2
(
6.75
cm
)
(
3
cm
)
≈
6.07
cm
2
Area of triangle=
2
1
(half chord)(radius)=
2
1
(
6.75
cm)(3 cm)≈6.07 cm
2
Therefore, the area of the segment is:
Area of segment
=
Area of sector
−
Area of triangle
=
9
4
�
cm
2
−
6.07
cm
2
≈
2.55
cm
2
Area of segment=Area of sector−Area of triangle=
4
9
π cm
2
−6.07 cm
2
≈2.55 cm
2
So, the area of the segment is approximately 2.55 square centimeters.