Chapter 5.5, Problem 36E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

# In regular hexagon A B C D E F , the length of A B is x centimeters. In terms of x find the length of a) diagonal B F ¯ . b) diagonal C F ¯ .

To determine

a) To find:

The exact length of the diagonal BF¯ of the regular hexagon ABCDEF and with AB=x cm.

Explanation

Approach:

For a right triangle, for which the measure of the interior angles 30Â°, 60Â°, and 90Â°; if â€˜aâ€™ is the length of measure of the shorter leg; opposite to the angle 30Â°, then the length of the other two sides is given by

Length of the longer leg (opposite to 60Â°) =a3

Length of the hypotenuse (opposite to 90Â°)=2a.

In general

Length of the longer leg =3Ã— (Length of the shorter leg)

Length of the hypotenuse =2Ã— (Length of the shorter leg)

Calculation:

Given,

A regular hexagon ABCDEF with the diagonal BF and of one of he face length measures AB=x cm.

Since, we have a regular hexagon, all the six faces are congruent and also all the six face angles,

AB=BC=CD=DE=EF=FA=x cm.

âˆ A=âˆ B=âˆ C=âˆ D=âˆ E=âˆ F=n-2Â·180Â°n

=6-2Â·180Â°6

=4Â·180Â°6

=4Â·30Â°

=120Â°

Thus, the measure of the length of each face of the polygon is x cm.

The measure of the each face angle of the polygon is 120Â°.

Here, the diagonal BF is perpendicular to the sides BC and EF of the hexagon.

Thus, the angles âˆ B and âˆ F of the âˆ†ABF is given by

âˆ B=âˆ F=120Â°-90Â°

âˆ B=âˆ F=30Â°

Consider the âˆ†ABF of the hexagon shown above and draw the auxiliary line APÂ¯âŠ¥BFÂ¯, which will bisect both the angle âˆ A of the triangle and also the base BF of the âˆ† as such ABF is an isosceles triangle with ABÂ¯â‰…AFÂ¯

To determine

b) To find:

The exact length of the diagonal CF¯ of the regular hexagon ABCDEF and with AB=x cm.

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