A Square ABCD inscribed in a circle of radius 4cm. What is the minor segment cut off by the chord AB
When you ask, "What is the minor segment cut off by the chord AB", I will assume
you meant its area.
sketch the square in the circle.
draw its two diagonals
area of the sector with arc AB = (1/4)π(4^2) = 4π square units
area of the right-angled triangle with AB as its hypotenuse
= (1/2)(4)(4) = 8 square units
So area of segment formed = 4π - 8 square units= appr .....
or , even easier way:
Area of circle = 16π
area of square: let each side of the square be x
x^2 + x^2 = 8^2
x^2 = 32
area = 32
area of one of the 4 segments = (1/4)(16π-32) = 4π - 8
I found it very helpful !
It's good and helpful but it's not more detail so u better make it better website
Thanks
Why did the circle go see a therapist? Because it had major issues with minor segments! In this case, the minor segment cut off by the chord AB can be found by subtracting the area of the triangle ABO (where O is the center of the circle) from the area of the sector ABO.
To find the minor segment cut off by the chord AB, we need to understand the concept of circular segments.
First, let's draw the given square ABCD inscribed in a circle:
```
B _______ A
| |
| |
| |
D|_______|C
```
Since the square is inscribed in a circle, each corner of the square touches the circumference of the circle. Thus, the chord AB is a diagonal of the square.
To determine the minor segment cut off by chord AB, we need to find the area of this segment. Here's how you can calculate it:
1. Determine the central angle:
Since the chord AB is a diagonal of the square, it divides the circle into two equal parts.
The central angle formed by the chord AB can be calculated as follows:
Central angle = 360 degrees / (number of sides of the polygon)
In this case, since we have a square (4 sides), the central angle is:
Central angle = 360 degrees / 4 = 90 degrees
2. Calculate the area of the sector:
The sector is the portion of the circle enclosed by the central angle. To calculate the area of the sector, use the formula:
Area of sector = (central angle in degrees / 360 degrees) * π * r^2
Where r is the radius of the circle.
In this case, r = 4 cm, and the central angle is 90 degrees, so the area of the sector is:
Area of sector = (90 degrees / 360 degrees) * π * 4^2 = π cm^2
3. Find the area of the triangle:
The triangle formed by chord AB and the radii drawn from the center to the endpoints of AB is an isosceles right triangle.
Since the radius is 4 cm, the length of the legs of the isosceles right triangle is 4 cm.
The area of the triangle can be calculated using the formula:
Area of triangle = (base * height) / 2
In this case, the base and height of the triangle are both 4 cm, so the area of the triangle is:
Area of triangle = (4 cm * 4 cm) / 2 = 8 cm^2
4. Calculate the area of the minor segment:
Finally, to find the area of the minor segment cut off by chord AB, subtract the area of the triangle from the area of the sector:
Area of minor segment = Area of sector - Area of triangle
= π cm^2 - 8 cm^2
Therefore, the area of the minor segment cut off by the chord AB is (π - 8) cm^2.