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Math Expert V
Joined: 02 Sep 2009
Posts: 57244
A semicircle with area of xπ is marked by seven points equally spaced  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 41% (03:12) correct 59% (03:07) wrong based on 75 sessions

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A semicircle with area of $$x \pi$$ is marked by seven points equally spaced along the half arc of the semicircle, such that two of the seven points form the endpoints of the diameter. What is the probability of forming a triangle with an area less than x from the total number of triangles formed by combining two of the seven points and the center of the diameter?

A. 4/5
B. 6/7
C. 17/21
D. 19/21
E. 31/35

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Math Expert V
Joined: 02 Aug 2009
Posts: 7754
A semicircle with area of xπ is marked by seven points equally spaced  [#permalink]

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5
2
Bunuel wrote:
A semicircle with area of $$x \pi$$ is marked by seven points equally spaced along the half arc of the semicircle, such that two of the seven points form the endpoints of the diameter. What is the probability of forming a triangle with an area less than x from the total number of triangles formed by combining two of the seven points and the center of the diameter?

A. 4/5
B. 6/7
C. 17/21
D. 19/21
E. 31/35

HI,

Surely a 700 level Q...
BUT the choices can make it a sub 600 level Q...
if you know this much that the total triangle possible is 7C2-1=20

rest alll choices have multiple of 7 in denominator which is not possible..
ONLY A is left

firstly semi circle has a area of $$x \pi$$..
so $$\frac{\pi*r^2}{2}=x\pi.......r=\sqrt{2x}$$
so when we make 7 points in the way it has been described, there are 6 equal segments which will have area of $$\frac{x \pi}{6}$$..LESS than $$x\pi$$
here central angle at centre is 180/6=30..
How many triangles ? 6

Now when we take two such pieces, the centre angle becomes 60 and other two sides are radius so it becomes EQUILATERAL triangle with each side $$\sqrt{2x}$$..
Area = $$\sqrt{3}/4*a^2=\sqrt{3}/4*\sqrt{2x}^2$$=$$\sqrt{3}/2*x$$, which is less than x.
How many such triangles? when 1 and 3 is choosen OR 2 and 4 OR 3 and 5 OR 4 and 6 OR 5 and 7------- 5 triangles..

Next when you choose three segments that is 30*3=90, it becomes right angled triangle with sides $$\sqrt{2x}$$..
area = $$\frac{1}{2}*\sqrt{2x}^2=x$$ which is NOT less than x.
How many such triangles ? 1-4, 2-5,3-6,4-7 points -----4 such triangles possible

After this the central angle will become >90 and so area will again start reducing or will become less than x

so total triangles with area equal to x is 4
total triangles possible = 7C2=$$\frac{7!}{5!2!}$$=21
..
but it includes the two points 1 and 7 where these three points form a straight line DIAMETER.. so 21-1=20
And triangles with area LESS than x is 20-4=16

probability = $$\frac{16}{20}=\frac{4}{5}$$

A
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Math Expert V
Joined: 02 Sep 2009
Posts: 57244
Re: A semicircle with area of xπ is marked by seven points equally spaced  [#permalink]

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chetan2u wrote:
Bunuel wrote:
A semicircle with area of $$x \pi$$ is marked by seven points equally spaced along the half arc of the semicircle, such that two of the seven points form the endpoints of the diameter. What is the probability of forming a triangle with an area less than x from the total number of triangles formed by combining two of the seven points and the center of the diameter?

A. 4/5
B. 6/7
C. 17/21
D. 19/21
E. 31/35

HI,

Bunuel, pl relook into the choices given..

firstly semi circle has a area of $$x \pi$$..
so $$\frac{\pi*r^2}{2}=x\pi.......r=\sqrt{2x}$$
so when we make 7 points in the way it has been described, there are 6 equal segments which will have area of $$\frac{x \pi}{6}$$..LESS than $$x\pi$$
here central angle at centre is 180/6=30..
How many triangles ? 6

Now when we take two such pieces, the centre angle becomes 60 and other two sides are radius so it becomes EQUILATERAL triangle with each side $$\sqrt{2x}$$..
Area = $$\frac{\sqrt{3[}{square_root]/4}*a^2=\frac{[square_root]3[}{square_root]/4}*[square_root]2x}^2=\frac{\sqrt{3[}{square_root]/2}*x$$, which is less than x.
How many such triangles? when 1 and 3 is choosen OR 2 and 4 OR 3 and 5 OR 4 and 6 OR 5 and 7------- 5 triangles..

Next when you choose three segments that is 30*3=90, it becomes right angled triangle with sides $$[square_root]2x}$$..
area = $$\frac{1}{2}*\sqrt{2x}^2=x$$ which is NOT less than x.
From here on any triangle with three segments or MORE will have AREA equal to or greater than x..

so total triangles with area less than x is 6+5=11
total triangles possible = 7C2=$$\frac{7!}{5!2!}$$=21
..
but it includes the two points 1 and 7 where these three points form a straight line DIAMETER.. so 21-1=20

probability = $$\frac{11}{20}$$

Checked. The options are copied as they show up in the source.
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Math Expert V
Joined: 02 Aug 2009
Posts: 7754
A semicircle with area of xπ is marked by seven points equally spaced  [#permalink]

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Bunuel wrote:
chetan2u wrote:
Bunuel wrote:
A semicircle with area of $$x \pi$$ is marked by seven points equally spaced along the half arc of the semicircle, such that two of the seven points form the endpoints of the diameter. What is the probability of forming a triangle with an area less than x from the total number of triangles formed by combining two of the seven points and the center of the diameter?

A. 4/5
B. 6/7
C. 17/21
D. 19/21
E. 31/35

HI,

Bunuel, pl relook into the choices given..

firstly semi circle has a area of $$x \pi$$..
so $$\frac{\pi*r^2}{2}=x\pi.......r=\sqrt{2x}$$
so when we make 7 points in the way it has been described, there are 6 equal segments which will have area of $$\frac{x \pi}{6}$$..LESS than $$x\pi$$
here central angle at centre is 180/6=30..
How many triangles ? 6

Now when we take two such pieces, the centre angle becomes 60 and other two sides are radius so it becomes EQUILATERAL triangle with each side $$\sqrt{2x}$$..
Area = $$\frac{\sqrt{3[}{square_root]/4}*a^2=\frac{[square_root]3[}{square_root]/4}*[square_root]2x}^2=\frac{\sqrt{3[}{square_root]/2}*x$$, which is less than x.
How many such triangles? when 1 and 3 is choosen OR 2 and 4 OR 3 and 5 OR 4 and 6 OR 5 and 7------- 5 triangles..

Next when you choose three segments that is 30*3=90, it becomes right angled triangle with sides $$[square_root]2x}$$..
area = $$\frac{1}{2}*\sqrt{2x}^2=x$$ which is NOT less than x.
From here on any triangle with three segments or MORE will have AREA equal to or greater than x..

so total triangles with area less than x is 6+5=11
total triangles possible = 7C2=$$\frac{7!}{5!2!}$$=21
..
but it includes the two points 1 and 7 where these three points form a straight line DIAMETER.. so 21-1=20

probability = $$\frac{11}{20}$$

Checked. The options are copied as they show up in the source.

Agreed Bunuel but it is surely 700 level Q
it skipped my mind that as the angle increases from 90 the area will again start reducing, so all other triangles will also be LESS than x
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Joined: 22 Aug 2017
Posts: 3
Re: A semicircle with area of xπ is marked by seven points equally spaced  [#permalink]

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@Brunuel,@
Please explain this question with a diagram. Thank You Re: A semicircle with area of xπ is marked by seven points equally spaced   [#permalink] 09 Aug 2018, 08:50
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# A semicircle with area of xπ is marked by seven points equally spaced

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