CAT Practice : Geometry-Triangles

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Hexagon inscribed inside circle

Hexagon inscribed inside circle

    Q.13: ABCDEF is a regular hexagon inscribed inside a circle. If the shortest diagonal of the hexagon is of length 3 units, what is the area of the shaded region.
    1. 1/6(3 − (93)/2)
    2. 1/6(2 − (63)/2)
    3. 1/6(3 − (83)/2)
    4. 1/6(6 − (153)/2)


  • Correct Answer
    Choice (A). 1/6(3 − (93)/2)

Explanatory Answer

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Detailed Solution

Let side of regular hexagon be a.

The shortest diagonal will be of length a3. Why?

A regular hexagon is just 6 equilateral triangles around a point. The shortest diagonal is FD.

FD = FP + PD

FOE is equilateral and so is EOD.

Diagonal FD can be broken as FP + PD, both of which are altitude of equilateral s.

FP = (3a)/2

FD = 3 a = shortest diagonal

The question tells us that the shortest diagonal measures 3 cm.

3 a = 3 => a = 3

Radius of circle = 3

Area of hexagon = (√3 a 2)/4 x 6

Area of circle – area of hexagon = π (√3)2 − √3/4 x (√3)2 x 6

= 3π − (9√3)/2

Area of shaded region = 1/(6 ) (area(circle) – area(hexagon))

= 1/(6 )(3π − (9√3)/2)

Answer choice (a).

Correct Answer: 1/6(3 − (93)/2)

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Geometry is probably the most vital topic as far as CAT preparation is concerned. Geometry sets the stage for Trigonometry, Cogeo and Mensuration as well.