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Difficulty: HardPythagorean Theorem and Special Right Triangles

A regular hexagon ABCDEFABCDEF has a side length of 88 inches. Point MM lies on side CDCD such that the length of segment CMCM is 22 inches. What is the length, in inches, of segment AMAM?

Answer: 14 inches

Answer

14
The correct answer is 14 because △ACM\triangle ACM is a right triangle with legs AC=83AC = 8\sqrt{3} and CM=2CM = 2. Applying the Pythagorean Theorem yields AM2=(83)2+22=192+4=196AM^2 = (8\sqrt{3})^2 + 2^2 = 192 + 4 = 196, so AM=196=14AM = \sqrt{196} = 14.

Step-by-Step Solution

1
Find the properties of the regular hexagon and the diagonal ACAC.
The interior angle at vertex BB is 120∘120^\circ. Since AB=BC=8AB = BC = 8, the triangle △ABC\triangle ABC is an isosceles triangle with angles ∠BAC=∠BCA=30∘\angle BAC = \angle BCA = 30^\circ. Using the properties of 30∘30^\circ-60∘60^\circ-90∘90^\circ triangles, the diagonal length is AC=83AC = 8\sqrt{3}.
To find the length of the leg ACAC for the right triangle △ACM\triangle ACM.
2
Determine the angle ∠ACD\angle ACD to show △ACM\triangle ACM is a right triangle.
Since the interior angle ∠BCD=120∘\angle BCD = 120^\circ and ∠BCA=30∘\angle BCA = 30^\circ, the remaining angle is ∠ACD=120∘−30∘=90∘\angle ACD = 120^\circ - 30^\circ = 90^\circ. Thus, △ACM\triangle ACM is a right triangle with the right angle at vertex CC.
To establish the right-angle relationship between the legs ACAC and CMCM.
3
Apply the Pythagorean Theorem to calculate the hypotenuse AMAM.
AM2=AC2+CM2=(83)2+22=192+4=196AM^2 = AC^2 + CM^2 = (8\sqrt{3})^2 + 2^2 = 192 + 4 = 196. Taking the square root gives AM=14AM = 14.
To find the final length of segment AMAM.

Key Concept

Using properties of regular hexagons, special right triangles, and the Pythagorean Theorem to find lengths in multi-step plane geometry configurations.
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Topics

Pythagorean Theorem and Special Right Triangles

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