Find the area of the segment cut off by a chord of length 10cm from a circle radius 13 cm.
From your sketch, join the end of the chord to the centre, drop a height from the centre to the chord.
You have a right-angled triangle, with 2 sides known, base is 5, hypotenuse is 13.
By Pythagoras, find the height
Find the central angle using basic trig.
find the circumference of the whole circle, then by a ratio , find the area of the sector with that central angle
Find the area of the triangle with your chord as base.
Subtract the two, to get the area of the segment.
There are formulas for the area of the segment, but the above makes you understand it.
Why did the chord go to the doctor? Because it wanted to understand its "length" condition!
But seriously, let's get down to business. To find the area of the segment cut off by the chord, we need to subtract the area of the triangle formed by the chord and the circle from the area of the sector formed by the same chord.
First, let's find the angle corresponding to the given chord. We can use the formula:
θ = 2 * arcsin (c / 2r)
Where c is the length of the chord (10 cm) and r is the radius of the circle (13 cm).
θ = 2 * arcsin (10 / 26)
Once we have the angle, we can find the area of the sector by using the formula:
A_sector = (θ / 360) * π * r²
And the area of the triangle can be found using the formula:
A_triangle = (1/2) * b * h
Where b is the base of the triangle (the length of the chord, 10 cm) and h is the height, which can be found by:
h = r - sqrt(r² - (c/2)²)
Now, the area of the segment is simply the difference between the area of the sector and the area of the triangle.
I hope you enjoyed this mathematical circus act!
To find the area of the segment cut off by a chord of length 10 cm from a circle with radius 13 cm, you can follow these steps:
Step 1: Find the central angle
The central angle of the chord can be found using the formula:
Central angle = 2 * arcsin(chord length / (2 * radius))
Substituting the values, we get:
Central angle = 2 * arcsin(10 / (2 * 13))
Step 2: Find the area of the sector
The area of the sector can be found using the formula:
Area of sector = (central angle / 360) * π * (radius^2)
Substituting the values, we get:
Area of sector = (central angle / 360) * π * (13^2)
Step 3: Find the area of the triangle
The area of the triangle can be found using the formula:
Area of triangle = (chord length / 2) * height
We can find the height using the formula:
height = radius * cos(central angle/2)
Substituting the values, we get:
Area of triangle = (10 / 2) * (13 * cos(central angle/2))
Step 4: Find the area of the segment
The area of the segment can be found by subtracting the area of the triangle from the area of the sector:
Area of segment = Area of sector - Area of triangle
Now, plug in the values and calculate step by step to get the final answer.
To find the area of the segment cut off by a chord, you need to determine the area of the corresponding sector and subtract the area of the triangle formed by the chord.
Here's how you can solve it step-by-step:
1. Calculate the central angle:
- The length of the chord divides the circle into two segments.
- You can use the formula for the length of a chord in a circle, which is given by 2r*sin(θ/2), where r is the radius and θ is the central angle.
- In this case, the chord length is 10 cm and the radius is 13 cm.
- So, 10 = 2*13*sin(θ/2).
- Divide both sides of the equation by 26 to get sin(θ/2) = 10/26.
- Take the inverse sine (sin^(-1)) of both sides to find θ/2.
2. Calculate the central angle:
- Double the value of θ/2 to find the central angle θ.
3. Calculate the area of the corresponding sector:
- The area of a sector is given by (θ/360) * π * r^2, where r is the radius and θ is the central angle.
- In this case, the radius is 13 cm and the central angle is θ (calculated in step 2).
- So, the area of the sector = (θ/360) * π * 13^2.
4. Calculate the area of the triangle:
- The area of a triangle can be calculated using the formula (1/2) * b * h, where b is the length of the base (chord) and h is the perpendicular distance from the vertex to the base.
- In this case, the length of the chord (base) is given as 10 cm.
- To find the perpendicular distance, draw two radii from the center of the circle to the endpoints of the chord. This will form a right triangle. You can solve for the height using Pythagoras' theorem.
- The hypotenuse of the right triangle is 13 cm (the radius).
- The base of the right triangle is half the length of the chord, which is 5 cm.
- Using Pythagoras' theorem, you can calculate the height.
- Once you have the height, you can calculate the area of the triangle.
5. Calculate the area of the segment:
- Subtract the area of the triangle (calculated in step 4) from the area of the corresponding sector (calculated in step 3).
By following these steps, you can find the area of the segment cut off by a chord of length 10 cm from a circle with radius 13 cm.