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CAT 2024 Slot 3 — QA Question 11

PolygonsEasy

A regular octagon ABCDEFGH has sides of length 6 cm each. Then, the area, in sq. cm, of the square ACEG is

Answer & solution

  • A

    72(2 + √2)

  • B

    36(1 + √2)

  • 36(2 + √2)

  • D

    72(1 + √2)

Solution

Medium

In a regular octagon, vertices A, C, E, G (every other vertex) form a square. Find the diagonal ACAC of the octagon, which is the side of that square, then square it.

1

Length of the short diagonal AC. A and C are separated by one vertex (B). The interior angle of a regular octagon is 135∘135^\circ, so in triangle ABCABC with AB=BC=6AB=BC=6 and angle B=135∘B=135^\circ, apply the cosine rule.

AC2=AB2+BC2−2 AB⋅BCcos⁡135∘⇒ AC2=36+36−2(36) ⁣(−22)(cos⁡135∘=−22)⇒ AC2=72+362\begin{aligned} &AC^2=AB^2+BC^2-2\,AB\cdot BC\cos135^\circ\\ &\Rightarrow\ AC^2=36+36-2(36)\!\left(-\tfrac{\sqrt2}{2}\right)\quad\text{(}\cos135^\circ=-\tfrac{\sqrt2}{2}\text{)}\\ &\Rightarrow\ AC^2=72+36\sqrt2 \end{aligned}
2

Area of square ACEG. ACEGACEG is a square with side ACAC, so its area is AC2AC^2 directly.

[ ACEG ]=AC2=72+362(from step 1)⇒ [ ACEG ]=36(2+2)\begin{aligned} &[\,ACEG\,]=AC^2=72+36\sqrt2\quad\text{(from step 1)}\\ &\Rightarrow\ [\,ACEG\,]=36(2+\sqrt2) \end{aligned}
A C E G
36(2+2)36(2+\sqrt2)
CAT 2024 Slot 3 QA Q11: A regular octagon ABCDEFGH has sides of length 6 cm each. Then, the area, in sq. cm, of the square ACEG is — Solution | TheCATExam