Question

A circle is divided into even number of sectors with central angles in arithmetic sequence. The smallest sector is of central angle 8° and the largest sector is of central angle 52°.
a) find the sum of the measures of central angle of all sectors
b) Find the sum of measures of central angle of first and last sectors
c) find the number of sectors ​

Answer 1

Answer:

Solution

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A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B.

Sum of all central angles is 360

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Answer 2

Given:

Smallest sector angle$\[ = {8^ \circ }\]$

largest sector angle$\[ = {52^ \circ }\]$

Solution:

part a)

Know that the central angle of a circle is $\[{360^ \circ }\]$.

Understand that from the question that the circle is divided into sectors.

Therefore, the sum of the measures of central angle of all sectors $=$central angle of the circle.

Hence, the sum of the measures of central angle of all sectors is $\[{360^ \circ }\]$.

part b)

Find the sum of measures of central angle of first and last sectors.

$\[{8^ \circ } + {52^ \circ } = {60^ \circ }\]$

Hence,  the sum of measures of central angle of first and last sectors is $\[{60^ \circ }\]$.

part c)

Find the number of sectors in which the circle is divided.

$\[\begin{array}{l}{S_n} = \frac{n}{2}\left[ {a + l} \right]\\ \Rightarrow 360 = \frac{n}{2}\left[ {8 + 52} \right]\\ \Rightarrow 360 = \frac{n}{2}\left[ {60} \right]\\ \Rightarrow n = 12\end{array}\]$

Hence, the number of sectors is 12.