Question

If the length of a minor arc of a circle is $\dfrac{1}{4}$ of its circumference, then the angle subtended by the minor arc at the center will be:
A. ${30^ \circ }$
B. ${45^ \circ }$
C. ${90^ \circ }$
D. ${60^ \circ }$

Answer 1

Hint: We will first write the formulas of the circumference of a circle and length of arc of a circle with regard to its angle. Then we will equate both according to the given condition and thus, we will have our answer.

Complete step-by-step answer:
Let us first discuss the formula of the circumference of a circle.
We know that $C = 2\pi r$, where r is the radius of the circle.
Hence, now we have: \[l = 2\pi r \times \dfrac{\theta }{{{{360}^ \circ }}}\], where $l$ is the arc length, r is the radius and $\theta $ is the angle covered.
Here, in the question we are given that the length of a minor arc of a circle is $\dfrac{1}{4}$ of its circumference.
This implies that \[2\pi r \times \dfrac{\theta }{{{{360}^ \circ }}} = \dfrac{1}{4} \times 2\pi r\].
Now, putting the values, we will get:-
\[2\pi r \times \dfrac{\theta }{{{{360}^ \circ }}} = \dfrac{1}{4} \times 2\pi r\]
Removing $2\pi r$ by cutting it off on both the sides:-
\[\dfrac{\theta }{{{{360}^ \circ }}} = \dfrac{1}{4}\]
Hence, $\theta = {360^ \circ } \times \dfrac{1}{4}$
Solving it, we will get:-
$\theta = {90^ \circ }$
Hence the required angle is a right angle.

So, the correct answer is “Option C”.

Note: The circumference (meaning "all the way around") of a circle is the line that goes around the centre of the circle. Mathematicians use the letter C for the length of this line.
The students must notice that arc length is a part of the circumference that is the perimeter of the circle only.
Fun Facts:- A circle is the only one sided shape with an area.
When you divide a circle's circumference you get $\pi $. The best number on earth is 3.14159265358979. March 14 or 3/14 is celebrated as pi day because 3.14 are the first digits of pi. Math nerds around the world love celebrating this infinitely long, never-ending number.
The awesome word encyclopedia literally means "circle of learning".
A human has no instinctive sense of direction so if there are absolutely no navigational clues, we will naturally walk in Circles and Circles.