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CAT 2017 Slot 2 — QA Question 15

HexagonEasy

Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

Answer & solution

Correct answer: 3

  • A

    3√2

  • 3

  • C

    4

  • D

    √3

Solution

Easy

The diagonal ACAC skips one vertex of the regular hexagon. In triangle ABCABC the equal sides AB=BC=1AB=BC=1 meet at the interior angle 120∘120^\circ; the Law of Cosines gives AC2AC^2, which is exactly the area of a square on ACAC.

A B C D E F
1

Interior angle. Each interior angle of a regular hexagon is 120∘120^\circ, so ∠ABC=120∘\angle ABC = 120^\circ with AB=BC=1AB = BC = 1.

∠ABC=(6−2)×180∘6=120∘\begin{aligned} &\angle ABC = \frac{(6-2)\times 180^\circ}{6} = 120^\circ \end{aligned}
2

Length of AC (Law of Cosines). Apply it to triangle ABCABC with cos⁡120∘=−12\cos 120^\circ = -\tfrac12.

AC2=AB2+BC2−2 AB⋅BCcos⁡120∘⇒ AC2=1+1−2(1)(1) ⁣(−12)(from step 1)⇒ AC2=3\begin{aligned} &AC^2 = AB^2 + BC^2 - 2\,AB\cdot BC\cos 120^\circ\\ &\Rightarrow\ AC^2 = 1 + 1 - 2(1)(1)\!\left(-\tfrac12\right) \quad\text{(from step 1)}\\ &\Rightarrow\ AC^2 = 3 \end{aligned}
3

Area of the square. A square with side ACAC has area AC2AC^2, already computed.

Area=AC2=3(from step 2)\begin{aligned} &\text{Area} = AC^2 = 3 \quad\text{(from step 2)} \end{aligned}
Area=3 sq cm\text{Area} = 3 \text{ sq cm}

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CAT 2017 Slot 2 QA Q15: Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one — Solution | TheCATExam