The length of the apothem of the considered regular hexagon is given by: Option B: 15.6 cm (approx).
What is Pythagoras Theorem?
If ABC is a triangle with AC as the hypotenuse and angle B with 90 degrees then we have:
[tex]|AC|^2 = |AB|^2 + |BC|^2[/tex]
where |AB| = length of line segment AB. (AB and BC are rest of the two sides of that triangle ABC, AC being the hypotenuse).
Renaming the triangle made inside the regular hexagon, as shown below, we get:
|AB| = length of the apothem to be known
|AC| = length of the hypotenuse of ABC triangle = 18 cm
|BC| = half of the length of the sides of the considered regular hexagon = 18/2 = 9 cm
(it is because regular polygons have all side same and that apothem bisects the side in two parts, (provable by symmetry))
Thus, using the Pythagoras theorem for triangle ABC as angle B internally in triangle ABC is of right angle (90 degrees), we get:
[tex]|AC|^2 = |AB|^2 + |BC|^2\\\\18^2 = |AB|^2 = 9^2\\|AB|^2 = 18^2 - 9^2 = 324 - 81 = 243\\\\|AB| = \sqrt{243} = 9\sqrt{3} \approx 15.6 \: \rm cm[/tex]
(took positive sq. root only and not negative sq. root since |AB| is length of AB line segment and length is a non-negative quantity).
Thus, the length of the apothem of the considered regular hexagon is given by: Option B: 15.6 cm (approx).
Learn more about Pythagoras theorem here:
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