dtweah wrote:getso wrote:Triangle ABC is inscribed in a circle with radius of 1; triangle DEF is within the circle and of
the vertexes is the center of the circle.what is the greatest possible are of trianble ABC minus
the greatest possible area of triangle DEF?
The answer is ( 3*3^1/2-2)/4
Please help in solving the above problem.
Thanks
What is this?" triangle DEF is within the circle and of
the vertexes is the center". Please edit and state problem clearly.
The problem needs no edit work except writing vertices in place of vertexes, and it is stated clearly too.
What we need to do here is to maximize the area of both the triangles. The inscribed triangle will have the maximum area in case it is an equilateral one. Do a little construction and recall the medians' property of triangles (the medians of a triangle divide each other in the ratio 2:1, 2 parts towards the vertex and 1 part towards the side) to determine one altitude of the equilateral triangle as to be 3/2. We know that the area in this case will be given by (3/2) ^2/√3 = 3 √3/4; and the area of the other triangle whose one vertex is at the center of the circle will be maximum in case it is right angled at the center, and the maximum area will be ½ (1) (1) = ½.
Now, the difference = 3 √3/4 – ½ =
(3 √3 – 2)/4. Take my answer.
Attached is the figure to make the understanding easier.
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The mind is everything. What you think you become. -Lord Buddha
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
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