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YellowGold
ACF is a 30º-60º-90º triangle because of the following:
1) Based on a theorem, in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : √3

1 → short leg
2 → hypotenuse
√3 → long leg

Side length of the hexagon is the short leg of the triangle. It is 1.

r1 is the radius of the incircle in a regular hexagon. 2(r1) is the diameter of the incircle. It is also the hypotenuse of the right triangle. It is 2.

Using Pythagorean theorem.

a² + b² = c²
1² + b² = 2²
b² = 2² - 1²
b² = 4 - 1
b² = 3
√b² = √3
b = √3









Ver imagen YellowGold
Ver imagen YellowGold

To solve the problem we must know about Thales' theorem and a regular hexagon.

What is Thales' theorem?

As we know according to the Thales' theorem which states that if A, B, and C are distinct points on a circle where the line BC is a diameter, the angle BAC is a right angle.

ΔACF is a 30º-60º-90º triangle.

Using Thales' theorem

The line FC is the diameter of the circle and hypotenuse as well, so ∠FAC = 90.

Regular Hexagon

For a regular hexagon, each of its angles measures 120°. So, in the figure, ∠EFA = 120°, but also at the same time line FC is bisecting the  ∠EFA which means ∠CFA will be equal to 60° in ΔFAC.

Sum of all Angles

Further, the sum of all the angles of a triangle is always 180°.

Hence, ΔACF is a 30º-60º-90º triangle.

Learn more about Thales' theorem:

https://brainly.com/question/14417137

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