Question:hard

The figure is a regular hexagon ABCDEF (in order) with side '2a' cm, drawn with C at the top vertex, F at the bottom vertex, B and A forming the left vertical edge, and D and E forming the right vertical edge. A rectangle is drawn using side AB as its left edge, with its bottom edge AG horizontal (G lying on line AE) and its right edge GH vertical (parallel to ED, with H directly above G). Given AG = FG and ED || GH, what is the ratio of the area of the shaded region (the triangle below AG down to F, plus the strip to the right of GH up to the hexagon's right edge DE) to the area of the hexagon?

Show Hint

Place the hexagon on coordinates with side 2a; use AG = FG to pin down where G falls on line y = -a, then split the shaded area into a triangle and a rectangle.
Updated On: Jul 20, 2026
  • 1 : 2
  • 1 : 3
  • 1 : 4
  • 2 : 3
  • 2 : 5
Show Solution

The Correct Option is A

Solution and Explanation

Work with fractions of the hexagon's width instead of raw coordinates.

A regular hexagon of side $2a$ splits into a central rectangle of width $2\sqrt3 a$ and height $2a$ (using the vertical edges AB and DE), plus two end triangles (cap at C, cap at F), each of area $\sqrt3 a^2$. The central rectangle has area $2\sqrt3a \times 2a = 4\sqrt3 a^2$, and together with the two caps ($\sqrt3a^2$ each) the total is $4\sqrt3a^2+\sqrt3a^2+\sqrt3a^2=6\sqrt3a^2$, matching the standard hexagon area formula $\frac{3\sqrt3}{2}(2a)^2$.

Placing A at the origin, F sits at $(\sqrt3a,-a)$ measuring along the bottom cap, and setting up $AG=FG$ for $G=(g,0)$ on line AE gives, after squaring and simplifying:
$g^2=(g-\sqrt3a)^2+a^2 \Rightarrow 0=-2\sqrt3ag+4a^2 \Rightarrow g=\dfrac{2a}{\sqrt3}$

This is exactly $\dfrac13$ of the full base $AE=2\sqrt3a$, so G splits AE in the ratio $1:2$ (AG : GE $= \tfrac{2a}{\sqrt3} : \tfrac{4a}{\sqrt3} = 1:2$).

Because the central rectangle splits into ABHG and GHDE in the same width ratio $1:2$, ABHG $=\tfrac13(4\sqrt3a^2)=\tfrac{4\sqrt3}{3}a^2$ and GHDE $=\tfrac23(4\sqrt3a^2)=\tfrac{8\sqrt3}{3}a^2$.

The bottom cap ($\sqrt3a^2$) splits at G in the same $1:2$ ratio (same base ratio, same apex F), so triangle AGF $=\tfrac13\sqrt3a^2=\tfrac{\sqrt3}{3}a^2$.

Shaded $=$ GHDE $+$ AGF $=\tfrac{8\sqrt3}{3}a^2+\tfrac{\sqrt3}{3}a^2=3\sqrt3a^2$, and hexagon $=6\sqrt3a^2$.

\[\boxed{Shaded : Hexagon = 1:2}\]

This matches the coordinate-geometry method exactly; the provided answer key (1:4) could not be reproduced.
Was this answer helpful?
0

Questions Asked in IBSAT exam

Although cynics may like to see the government’s policy for women in terms of the party’s internal power struggles, it will nevertheless be churlish to deny that it represents a pioneering effort aimed at bringing about sweeping social reforms. In its language, scope and strategies, the policy document displays a degree of understanding of women’s needs that is uncommon in government pronouncements. This is due in large part to the participatory process that marked its formulation, seeking the active involvement right from the start of women’s groups, academic institutions and non-government organizations with grass roots experience.
The result is not just a lofty declaration of principles but a blueprint for a practical programme of action. The policy delineates a series of concrete measures to accord women a decision-making role in the political domain and greater control over their economic status. Of especially far-reaching impact are the devolution of control of economic infrastructure to women, notably at the gram panchayat level, and the amendments proposed in the Hindu Succession Act of 1956 to give women coparcenary rights.
 An enlightened aspect of the policy is its recognition that actual change in the status of women cannot be brought about by the mere enactment of socially progressive legislation. Accordingly, it focuses on reorienting development programmes and sensitizing administrations to address specific situations as, for instance, the growing number of households headed by women, which is a consequence of rural-urban migration. The proposal to create an equal-opportunity police force and give women greater control of police stations is an acknowledgement of the biases and callousness displayed by the generally all-male lawenforcement authorities in cases of dowry and domestic violence. While the mere enunciation of such a policy has the salutary effect of sensitizing the administration as a whole, it does not make the task of its implementation any easier. 
This is because the changes it envisages in the political and economic status of women strike at the root of power structures in society and the basis of man woman relationships. There is also the danger that reservation for women in public life, while necessary for their greater visibility, could lapse into tokenism or become a tool in the hands of vote seeking politicians. Much will depend on the dissemination of the policy and the ability of elected representatives and government agencies to reorder their priorities.
  • IBSAT
  • Reading Comprehension
View Solution
Generally, people experience stress in their day-to-day lives, but more than thirty million Americans suffer from something more intense than that. Anxiety disorders are the second-most-common mental health problem in the country, and they can be paralyzing. Sometimes, day-to-day stress and anxiety are hard to tell apart, but the easiest way to distinguish them is that stress is brought on by actual events, and then dissipates, whereas anxiety is a more pervasive worry, that often attaches itself to specific areas of your life, like your relationship, job or health, says psychologist Terry Mooney.
This anxiety does not dissipate, and in fact, it can increase to the level where it begins to change your behavior. That’s when it’s characterized as an anxiety disorder. Anxiety can keep you safe, helping you recognize danger, and cope with it, says Mooney. But if you begin to see danger lurking around every corner, or worry over and over again about the same events, then you might be dealing with something more substantial, like an anxiety disorder. 
People with anxiety disorders often begin to avoid activities or circumstances that make them anxious, said John Forsyth, associate professor of Psychology at the State University of New York at Albany and director of the Anxiety Disorders Research Programme at the university. People may stop driving, or stop going to parties. They may stop travelling or even avoid leaving the house. It’s this curtailing of activities that causes the suffering, Forsyth said, making you feel that ‘life is shrinking around you.’
 Anxiety disorders come in a variety of forms and manifest themselves in different ways. Generalized anxiety disorder is characterized by ongoing worry about everyday tasks, even when there is no clear reason to worry. People with social anxiety disorder experience intense worry over social interactions, and often feel judged by people or worry that they will embarrass themselves. Post-traumatic stress disorder, which is characterized by people re-living a frightening event over and over again, is also considered an anxiety disorder. Obsessive-compulsive disorder is when people take on obsessive rituals that help them maintain the illusion of control. Often repetitive in nature, obsessive compulsive rituals can include cleaning, checking and rechecking something, counting or endlessly reviewing conversations in their mind. Treatment is available for anxiety disorders. People can use a variety of approaches, including therapy, medication and exercise.
  • IBSAT
  • Reading Comprehension
View Solution
I am the family face; flesh perishes. I live on, projecting trait and trace through time to times anon, and leaping from place to place over oblivion.
So wrote Thomas Hardy in his poem, Heredity, describing direct descent of life from one generation to the next. Indeed, the poem reflects the DNA in our genome. Dr.Drew Endy of MIT quoted this when he described how people at the J. Craig Venter Institute (JCVI) bypass nature’s constraint of direct descent. Scientists there have used chemistry and biochemistry to produce the first synthetic genome in the laboratory. They chemically synthesized many fragments of the DNA, encoding the 582,970-units-long genome of a bacterium called Mycoplasma genitalium. Next, they assembled these fragments in perfect order to generate the genome of the bacterium.
The DNA sequence of the synthetic one was confirmed to be identical to the natural one. While the DNA pieces were synthesized chemically, the stitching together was done using the biochemical machinery of a host cell. About 100 pieces of the genome, each 5000-7000 units long in DNA sequence, were first joined to produce 25 sub-assemblies, each about 24000 base pairs long. These were then introduced into the bacterium E. coli to produce sufficient DNA for the next steps. Next, they repeated the procedure to generate large fragments comprising l/4th of the whole genome of M.genitalium.
Now, they used the clever trick of exploiting the process called homologous recombination. This is a basic essential process in every cell, which physically rearranges the two strands of DN The JCVI researchers inserted the synthesized DNA fragments into yeast and utilized its homologous recombination ability to generate the whole 580,000 long genome of M.genitalium in one step. 
This is clearly a landmark work that leads into the brave new world of synthesizing life itself in the laboratory. It was hardly 200 years ago when Friedrich Wohler synthesized urea, an organic molecule, in the chemical laboratory, thus throwing out the notion of ‘vital forces’ involved in the components of living organisms. What is the next step, making life itself in the lab, bypassing nature? With single cell organisms like M.genitalium, it might not be far away. It is now possible in the lab to do so, by inserting the genome into a ‘host’ cell and asking the latter to make the bacterium of your choice. If only we find a way to insert the bacterial genome into this proto-cell, and somehow trigger it to make the bacterium itself! We would have chemically created life in the la This is not a pipedream; JCVI scientists are already on the job, and my bet is they will do it within a few years. 
This surely raises ethical questions, a matter that JCVI is keenly aware of and is already engaged in with ethicists. Even their present work on M.genitalium was done with prior approval of ethical experts. But then, today it is M.genitalium, tomorrow it could be a more advanced, multi-cellular organism, and that could flummox even the ethicist. Assisted reproduction, which is the other side of the coin and truly a recently initiated technology, has become ethically and morally acceptable. Cloning of Dolly, the sheep, has not raised any outrage, but cloning a human certainly does.
  • IBSAT
  • Reading Comprehension
View Solution