Day 7 Problem

Triangle FAN is circumscribed in a circle G. Fang already solved for the following in the circle: NG measures 4 meters, FN measures 5 meters, and AN measures 6 meters. What should Fang's answer regarding the measure of arc FA be?

This problem was originally posted at @savvyplayer/mayday7math.

Answer

12.19 meters 🎯

Equivalent answers in terms of π (which will give an answer of approximately 3.88π) will also be considered correct.

Solution

In this approach, we get the measures of the two other intercepted arcs FN and NA and deduct them from the circumference to get the measure of arc FA.

The circumference of the circle is very easy to get, since we already have the circle's radius which is 4 meters. The circumference is 8π meters or approximately 25.1327 meters.

Triangle NGF has sides NG, FG, and NF, which are 4 meters, 4 meters, and 5 meters, respectively. We can get the measure of arc FN by using the Arc Cosine function together with the Law of Cosines formula. The Law of Cosines is c² = a² + b² - 2 * a * b * cos (C) where cos (C) = (a² + b² - c²) / (2 * a * b), which when inserted to the arccos function gives C = arccos [(a² + b² - c²) / (2 * a * b)] = arccos [(4² + 4² - 5²) / (2 * 4 * 4) ≈ 77.3643°.

Likewise as the last paragraph, triangle NGA has sides NG, FA, and NA, which are 4 meters, 4 meters, and 6 meters, respectively. Using the arccos function together with the Law of Cosines, we get C = arccos [(a² + b² - c²) / (2 * a * b)] = arccos [(4² + 4² - 6²) / (2 * 4 * 4) ≈ 97.1808°

To get the percentage of arc FA to be able to get its actual measure, we need to get the percentage of the circumference occupied by the other two arcs. The two other arcs occupy `(77.3643° + 97.1808°) / 360° ≈ 48.485%. Therefore, arc FA is approximately 51.515% of the circumference of the circle, which has a result of approximately 12.1855 meters.

Winner: @minus-pi 🏅

1 HIVE has been sent to @minus-pi's Hive account. 💰

Despite my Math challenge problems for Day 6 and Day 7 more difficult than usual, @minus-pi was able to solve them correctly!

 
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