In a circle, two arcs of unequal length subtend central angles in the ratio 5 : 3. If the smaller central angle is 45 degrees, what is the measure of the larger central angle?

Introduction / Context:

This geometry question tests your understanding of the direct proportionality between arc lengths and the central angles that subtend them. It is a typical quick question in circle geometry for aptitude exams.

Given Data / Assumptions:

• Two arcs of a circle subtend central angles in the ratio 5 : 3.
• The smaller central angle is 45 degrees.
• We need to find the larger central angle.
• The ratio 5 : 3 corresponds to the ratio of the central angles.

Concept / Approach:

In a circle, the length of an arc is directly proportional to the central angle subtending that arc. Therefore, if the central angles are in the ratio 5 : 3, each angle can be written as 5k and 3k degrees. We are told that the smaller angle is 45 degrees, so we can solve for k and then find the larger angle.

Step-by-Step Solution:

Verification / Alternative check:

Check the ratio: larger : smaller = 75 : 45. Divide both by 15 to get 5 : 3, which matches the given ratio of central angles, confirming the correctness.

Why Other Options Are Wrong:

The values 72 degrees, 60 degrees, 78 degrees, and 90 degrees do not maintain the required ratio of 5 : 3 with 45 degrees. For example, 72 : 45 simplifies to 8 : 5, which is not 5 : 3.

Common Pitfalls:

Some learners mistakenly take 45 degrees as the larger angle and 5k as 45, which would give k = 9 and lead to a smaller angle of 27 degrees, contradicting the question. It is critical to match the correct part of the ratio with the given angle.

Final Answer:

The measure of the larger central angle is 75°.