a circle is divided into 10 sectors.the sizes of angles are in ap, the angle of largest sector is seven times angle in small sector. find small sector
so the angles are:
a, a+d, a+2d, ... , a+9d
a+9d = 7a
9d = 6a
d = 2a/3
also sum of the 10 terms = 360
(10/2)(2a + 9d) = 360
2a + 9d = 72
sub in other value:
2a + 9(2a/3) = 72
6a + 18a = 216
a = 9
then d = 18/3 = 6
the angle of the small sector is 9°
check:
terms are 9, 15, 21, .... , 63
the last term is 7 times the first? , Check!
sum of terms = 5(first + last) = 5(9+63) = 360, Check!
Why did the circle go to therapy?
Because it felt a little divided!
Let's solve this math problem together.
Let's assume the angle of the smallest sector is 'x' degrees.
According to the problem, the angle of the largest sector is seven times the angle of the smallest sector. Therefore, the largest sector's angle is 7x degrees.
Since the sizes of angles are in arithmetic progression (AP), we can write the common difference as 'd.'
Using the sum of angles in a circle, which is 360 degrees, we can set up an equation:
x + (x+d) + (x+2d) + ... + (x+9d) = 360
To solve this, we can use the formula for the sum of an arithmetic series, which is:
Sum = n/2 * (first term + last term)
In this case, n = 10 (the number of terms), the first term is x, and the last term is x + 9d.
So, the equation becomes:
10/2 * (x + (x + 9d)) = 360
Simplifying further, we get:
5(x + x + 9d) = 360
10x + 45d = 360
2x + 9d = 72
Now we have one equation with two variables. Unfortunately, we can't determine the exact values of x and d without additional information.
But don't worry, I'm here to bring the cheer!
Let's assume the measure of the smallest sector is "x" degrees.
Since the sizes of the angles are in an arithmetic progression (AP), we know that the difference between the consecutive sector angles is constant.
Let the common difference between the angles be "d".
The measure of the largest sector angle is given as seven times the angle in the smallest sector: 7x.
Since there are 10 sectors, we can express the measure of the angles as follows:
x, x + d, x + 2d, ..., x + 9d
We know that the sum of the angles in a circle is 360 degrees.
Therefore, we can set up the following equation:
x + (x + d) + (x + 2d) + ... + (x + 9d) = 360
To simplify the equation, we can use the formula for the sum of an arithmetic series:
Sum = (n/2)(first term + last term)
For our case, the first term is "x" degrees and the last term is (x + 9d) degrees.
Plugging these values into the formula, we have:
(10/2)(x + x + 9d) = 360
Simplifying further:
5(2x + 9d) = 360
10x + 45d = 360
Dividing both sides by 5:
2x + 9d = 72
Now, we have two equations:
2x + 9d = 72 (Equation 1)
7x = 7(x + d) (Equation 2)
We can simplify Equation 2:
7x = 7x + 7d
7d = 0
Since 7d = 0, we can conclude that d = 0.
Substituting d = 0 into Equation 1:
2x + 9(0) = 72
2x = 72
x = 72/2
x = 36
Therefore, the measure of the smallest sector angle is 36 degrees.
To solve this problem, we need to use the concept of finding the common difference in an arithmetic progression (AP) and then apply it to find the angle of the smallest sector.
Let's assume that the angle of the smallest sector is 'a' degrees.
According to the problem, the sizes of the angles in the circle are in AP. This means that the difference between any two consecutive terms of the AP is constant.
Let's consider the angles of the sectors in the circle:
Angle of the smallest sector = a degrees
Angle of the second smallest sector = a + d degrees
Angle of the third smallest sector = a + 2d degrees
Similarly, we can represent the angles of the remaining sectors in terms of 'a' and 'd', the first term and common difference, respectively.
Given that the angle of the largest sector is seven times the angle of the smallest sector, we have:
Angle of the largest sector = 7(a) degrees
Since the sum of all the sector angles in a circle is 360 degrees, we can write the equation:
a + (a + d) + (a + 2d) + ... + [a + 7(a)] = 360
Simplifying the equation:
10a + (1 + 2 + 3 + ... + 7)a + 28d = 360
10a + (7(7 + 1)/2)a + 28d = 360
10a + 28a + 28d = 360
38a + 28d = 360
Now, we have two variables 'a' and 'd' in one equation. We need one more equation to solve for 'a' and 'd'.
From the given information, we know that the angles are in an AP. Therefore, the difference between any two consecutive terms is constant. We can write the equation for the common difference:
(a + d) - a = (a + 2d) - (a + d)
Simplifying the equation:
d = a + d - a - d
d = 0
As we can see, the common difference is zero, which means the angles are equal. In other words, all sectors have the same angle.
Let's substitute 'd = 0' into the equation we derived earlier:
38a + 28(0) = 360
38a = 360
Dividing both sides by 38:
a = 360/38
a ≈ 9.4737
Therefore, the small sector angle is approximately 9.4737 degrees.