Question 4·Easy·Circles
In the circle shown, radii and form a right angle at the center . The circle has a circumference of 32.
What is the length of minor arc ?
For circle arc-length questions, first identify the central angle and express it as a fraction of (for example, a right angle is , which is of a circle). Then multiply that fraction by the given circumference to get the arc length, and only do as much calculation as needed—often you can see it’s a simple fraction of the total, like one-half or one-fourth, and compute it quickly without extra formulas.
Hints
Use the central angle
A full circle has . What is the measure of angle , and what fraction of is that?
Connect angle fraction to arc length
The arc length is the same fraction of the circumference as the central angle is of . Once you know the angle fraction, multiply it by the total circumference of 32.
Do the fraction multiplication
You should get a simple fraction like . Multiply that fraction by 32 to find the arc length.
Desmos Guide
Enter the arc length expression
In a new expression line, type (90/360)*32 or equivalently (1/4)*32 to represent the fraction of the circumference that matches a central angle.
Read the numerical result
Look at the value that Desmos displays for this expression; that number is the length of minor arc . Compare it to the answer choices.
Step-by-step Explanation
Relate the central angle to the whole circle
A full circle has . The problem says and form a right angle at the center, so the central angle is .
Find what fraction of the full circle this is:
So arc represents of the entire circle.
Connect arc length to circumference
The length of an arc is the same fraction of the circle’s circumference as its central angle is of .
So, if the full circumference is 32 and the arc is of the circle, then the arc length is:
Now compute this product.
Compute the arc length and match the choice
Calculate the product:
So the length of minor arc is 8, which corresponds to choice B.