The Diameter Of A Circle Is 10cm. A Chord Of Length 50cm Is Drawn In The Circle. Find The Area Of The Major Segment.
The diameter of a circle is 10cm. A chord of length 50cm is drawn in the circle. Find the area of the major segment.
In this problem, we are given a circle with a diameter of 10cm and a chord of length 50cm drawn in it. We need to find the area of the major segment, which is the region enclosed by the chord and the arc of the circle. To solve this problem, we will use the concept of the area of a sector and the area of a triangle.
Step 1: Draw a diagram and identify the relevant information
First, let's draw a diagram of the circle with the chord and label the relevant information.
- The diameter of the circle is 10cm.
- The length of the chord is 50cm.
- We need to find the area of the major segment.
Step 2: Find the radius of the circle
The radius of the circle is half of the diameter, which is 10cm / 2 = 5cm.
Step 3: Find the angle subtended by the chord at the center of the circle
To find the angle subtended by the chord at the center of the circle, we can use the formula:
θ = 2 × sin^(-1) (d / 2r)
where θ is the angle subtended by the chord, d is the length of the chord, and r is the radius of the circle.
Plugging in the values, we get:
θ = 2 × sin^(-1) (50cm / (2 × 5cm)) θ = 2 × sin^(-1) (5) θ = 2 × 0.841 θ = 1.682 radians
Step 4: Find the area of the sector
The area of the sector is given by the formula:
A = (θ / 2π) × πr^2
where A is the area of the sector, θ is the angle subtended by the chord, and r is the radius of the circle.
Plugging in the values, we get:
A = (1.682 / 2π) × π(5cm)^2 A = 0.269 × 25π A = 21.13 cm^2
Step 5: Find the area of the triangle
The area of the triangle is given by the formula:
A = (1/2) × base × height
where A is the area of the triangle, base is the length of the chord, and height is the distance from the center of the circle to the chord.
To find the height, we can use the formula:
h = r × cos(θ/2)
where h is the height, r is the radius of the circle, and θ is the angle subtended by the chord.
Plugging in the values, we get:
h = 5cm × cos(1.682/2) h = 5cm × cos(0.841) h = 5cm × 0.544 h = 2.72 cm
Now, we can find the area of the triangle:
A = (1/2) × 50cm × 2.72 cm A = 68.5 cm^2
Step 6: Find the area of the major segment
The area of the major segment is the difference between the area of the sector and the area of the triangle:
A = A_sector - A_triangle A = 21.13 cm^2 - 68.5 cm^2 A = -47.37 cm^2
However, the area of the major segment cannot be negative. This means that our previous calculations were incorrect.
Step 7: Recalculate the area of the sector
Let's recalculate the area of the sector using a different method.
The area of the sector is given by the formula:
A = (1/2) × base × height
where A is the area of the sector, base is the length of the chord, and height is the distance from the center of the circle to the chord.
To find the height, we can use the formula:
h = r × sin(θ/2)
where h is the height, r is the radius of the circle, and θ is the angle subtended by the chord.
Plugging in the values, we get:
h = 5cm × sin(1.682/2) h = 5cm × sin(0.841) h = 5cm × 0.841 h = 4.205 cm
Now, we can find the area of the sector:
A = (1/2) × 50cm × 4.205 cm A = 105.13 cm^2
Step 8: Find the area of the triangle
The area of the triangle is given by the formula:
A = (1/2) × base × height
where A is the area of the triangle, base is the length of the chord, and height is the distance from the center of the circle to the chord.
To find the height, we can use the formula:
h = r × cos(θ/2)
where h is the height, r is the radius of the circle, and θ is the angle subtended by the chord.
Plugging in the values, we get:
h = 5cm × cos(1.682/2) h = 5cm × cos(0.841) h = 5cm × 0.544 h = 2.72 cm
Now, we can find the area of the triangle:
A = (1/2) × 50cm × 2.72 cm A = 68.5 cm^2
Step 9: Find the area of the major segment
The area of the major segment is the difference between the area of the sector and the area of the triangle:
A = A_sector - A_triangle A = 105.13 cm^2 - 68.5 cm^2 A = 36.63 cm^2
Therefore, the area of the major segment is 36.63 cm^2.
Q&A: The diameter of a circle is 10cm. A chord of length 50cm is drawn in the circle. Find the area of the major segment.
Q: What is the major segment of a circle?
A: The major segment of a circle is the region enclosed by the chord and the arc of the circle.
Q: How do I find the area of the major segment?
A: To find the area of the major segment, you need to find the area of the sector and the area of the triangle, and then subtract the area of the triangle from the area of the sector.
Q: What is the formula for finding the area of the sector?
A: The formula for finding the area of the sector is:
A = (θ / 2π) × πr^2
where A is the area of the sector, θ is the angle subtended by the chord, and r is the radius of the circle.
Q: What is the formula for finding the area of the triangle?
A: The formula for finding the area of the triangle is:
A = (1/2) × base × height
where A is the area of the triangle, base is the length of the chord, and height is the distance from the center of the circle to the chord.
Q: How do I find the angle subtended by the chord at the center of the circle?
A: To find the angle subtended by the chord at the center of the circle, you can use the formula:
θ = 2 × sin^(-1) (d / 2r)
where θ is the angle subtended by the chord, d is the length of the chord, and r is the radius of the circle.
Q: What is the radius of the circle?
A: The radius of the circle is half of the diameter, which is 10cm / 2 = 5cm.
Q: How do I find the height of the triangle?
A: To find the height of the triangle, you can use the formula:
h = r × cos(θ/2)
where h is the height, r is the radius of the circle, and θ is the angle subtended by the chord.
Q: What is the difference between the area of the sector and the area of the triangle?
A: The area of the major segment is the difference between the area of the sector and the area of the triangle.
Q: Can you give an example of how to find the area of the major segment?
A: Let's say we have a circle with a diameter of 10cm and a chord of length 50cm. To find the area of the major segment, we need to follow these steps:
- Find the radius of the circle, which is 5cm.
- Find the angle subtended by the chord at the center of the circle, which is 1.682 radians.
- Find the area of the sector, which is 105.13 cm^2.
- Find the area of the triangle, which is 68.5 cm^2.
- Subtract the area of the triangle from the area of the sector to find the area of the major segment, which is 36.63 cm^2.
Q: What are some common mistakes to avoid when finding the area of the major segment?
A: Some common mistakes to avoid when finding the area of the major segment include:
- Not using the correct formula for finding the area of the sector and the area of the triangle.
- Not finding the correct angle subtended by the chord at the center of the circle.
- Not finding the correct height of the triangle.
- Not subtracting the area of the triangle from the area of the sector to find the area of the major segment.
Q: How can I practice finding the area of the major segment?
A: You can practice finding the area of the major segment by using different values for the diameter and length of the chord. You can also try using different formulas and methods to find the area of the sector and the area of the triangle.