Question

consider a regular hexagon inscribed in circle c with radius r. regular hexagons have six congruent sides and six congruent angles. in the figure, angle cab measures 60°. what is m∠acb? ° what is the length of segment ab? what is the perimeter of the hexagon? the perimeter of the hexagon is the circumference of the circle. the circumference of the circle is

Explanation:

First Sub - Question: Find \(m\angle ACB\)

Step 1: Recall properties of regular hexagon

In a regular hexagon inscribed in a circle, the central angle (angle subtended at the center of the circle) corresponding to each side is \(\frac{360^{\circ}}{6}=60^{\circ}\). Also, in \(\triangle ACB\), \(AC = BC=r\) (radii of the circle) and we know \(\angle CAB = 60^{\circ}\).

Step 2: Use triangle properties

Since \(AC = BC\) and \(\angle CAB=60^{\circ}\), by the property of equilateral triangles (if two sides are equal and one angle is \(60^{\circ}\), the triangle is equilateral), \(\triangle ACB\) is equilateral. So, \(\angle ACB = 60^{\circ}\).

Step 1: Analyze \(\triangle ACB\)

We know that \(\triangle ACB\) is equilateral (from the first sub - question). In an equilateral triangle, all sides are equal. Since \(AC = r\) (radius of the circle), \(AB=AC = r\).

Step 1: Recall the number of sides of a hexagon

A regular hexagon has 6 sides.

Step 2: Calculate the perimeter

Since each side of the hexagon (e.g., \(AB\)) has length \(r\) (from the second sub - question), the perimeter \(P=6\times r = 6r\).

Answer:

\(60\)

Second Sub - Question: Find the length of segment \(AB\)