Fiona must find the length indicated by the dotted line for the tiles she is installing. She knows that each polygon is a regular hexagon with a perimeter of 7. 5 in. 3 regular hexagons are connected at their sides. A dotted line is drawn through the middle of the 3 hexagons. The top image is a reflection of the bottom image. The dotted line is the length of one side and the distance from one point of the hexagon to the opposite point. What is the length of the dotted line? Round to the nearest hundredth. 2. 50 in. 3. 75 in. 5. 00 in. 6. 25 in.

A regular hexagon is a polygon with 6 sides whose all sides are of equal length, and all the angles are of equal measurement. The angle made by each side of the hexagon at the center of the hexagon is 60°.

The length of the dotted line is 3.75 in.

Explanation

In ΔAOB,

OA = OB = at the same distance from the center of the hexagon,

therefore, the triangle is an isosceles triangle.

∠OAB = ∠OBA

Sum of all angles of a triangle

The sum of all the angles of a triangle is equal to 180°.

∠OAB + ∠OBA + ∠AOB = 180°

2 ∠OAB + 60° = 180°; {∠OAB = ∠OBA}

2∠OAB = 180° - 60°

2∠OAB = 120°

∠OAB = 60°

∠OAB = ∠OBA = 60°

Thus, the ΔAOB is an equilateral triangle.

Therefore, OA = OB = AB = a.

Similarly, for the tiles, the dotted line will be equal to (a+a+a = 3a). where a is the side of the hexagon.

Side of Hexagon

Perimeter of the Hexagon = 6 x a

7.5 in = 6 x a

[tex]a =\dfrac{7.5}{6} = 1.25\ in[/tex]

Length of Dotted Line

Further, Dotted line = 3 x a

= 3 x 1.25

= 3.75 in

Hence, the length of the dotted line is 3.75 in.

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