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# In the figure above, the letters L, M, and N denote the areas of the

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Math Expert
Joined: 02 Sep 2009
Posts: 59125
In the figure above, the letters L, M, and N denote the areas of the  [#permalink]

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31 Oct 2018, 02:10
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Difficulty:

35% (medium)

Question Stats:

69% (01:56) correct 31% (02:16) wrong based on 46 sessions

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In the figure above, the letters L, M, and N denote the areas of the semicircular regions whose diameters are the sides of the triangle, as shown. What is the value of (L + M)/N ?

A. 1/2

B. √2/2

C. 1

D. π/(2√2)

E. 2√2

Attachment:

phd02.png [ 15.19 KiB | Viewed 869 times ]

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Re: In the figure above, the letters L, M, and N denote the areas of the  [#permalink]

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31 Oct 2018, 05:39
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Bunuel wrote:

In the figure above, the letters L, M, and N denote the areas of the semicircular regions whose diameters are the sides of the triangle, as shown. What is the value of (L + M)/N ?

A. 1/2

B. √2/2

C. 1

D. π/(2√2)

E. 2√2

Attachment:
phd02.png

Since it is a right triangle, assume that the sides are 6-8-10 so that radii of the semi circles are 3-4-5

The areas of semi circles (pi*r^2)/2 will be in the ratio of r^2 then i.e. 9:16:25

(L + M)/N = (9 + 16)/25 = 1
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Director
Joined: 19 Oct 2013
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Re: In the figure above, the letters L, M, and N denote the areas of the  [#permalink]

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31 Oct 2018, 02:13
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Hi Bunuel I don't see any figure shown, can you please check?
Math Expert
Joined: 02 Sep 2009
Posts: 59125
Re: In the figure above, the letters L, M, and N denote the areas of the  [#permalink]

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31 Oct 2018, 02:17
Salsanousi wrote:
Hi Bunuel I don't see any figure shown, can you please check?

_______________
Fixed. Thank you.
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In the figure above, the letters L, M, and N denote the areas of the  [#permalink]

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Updated on: 31 Oct 2018, 03:33
let the diameter of semicircle n be C

Using pythagorean theorem $$C^2 = A^2 + B^2$$

Area of semicircle $$= Pi * r^2 * \frac{1}{2} = Pi * \frac{D^2}{4} * \frac{1}{2} = Pi * \frac{D^2}{8}$$

From the above the area of semicircle $$N = Pi * \frac{C^2}{8}$$

Area of semicircle $$L = Pi * \frac{B^2}{8}$$

Area of semicircle $$M = Pi * \frac{A^2}{8}$$

Since B^2 + A^2 = C^2 then L+M = N

Added together $$\frac{L+M}{N}$$= $$(Pi * B^2/8 + Pi *A^2/8)/Pi * C^2/8 = 1$$

Originally posted by Salsanousi on 31 Oct 2018, 02:46.
Last edited by Salsanousi on 31 Oct 2018, 03:33, edited 1 time in total.
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Joined: 17 Nov 2016
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Re: In the figure above, the letters L, M, and N denote the areas of the  [#permalink]

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31 Oct 2018, 03:21
Salsanousi wrote:
let the diameter of semicircle n be C

Using pythagorean theorem $$C^2 = A^2 + B^2$$

Area of semicircle $$= Pi * r^2 * \frac{1}{2} = Pi * \frac{D^2}{4} * \frac{1}{2} = Pi * \frac{D^2}{8}$$
From the above the area of semicircle N = Pi * \frac{C^2}{8}

Area of semicircle $$L = Pi * \frac{B^2}{8}$$

Area of semicircle $$M = Pi * \frac{A^2}{8}$$

Since B^2 + A^2 = C^2 then L+M = N

Added together \frac{L+M}{N}= $$(Pi * B^2/8 + Pi *A^2/8)/Pi * C^2/8 = 1$$

Hi,
Can you please explain how you got L+M =N?

As per Pythagorean theorem, L^2+M^2=N^2 i understood but how did you get L+M = N

Thanks

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Re: In the figure above, the letters L, M, and N denote the areas of the  [#permalink]

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31 Oct 2018, 03:35
Hi shashaankbhat

You know we get Pi * B^2/8 + Pi * A^2/8 correct?

Now if we take Pi/8 as a common factor we have (B^2 + A^2) which equals C^2

So it becomes Pi/8 * C^2/ (Pi/8 * C^2) and this cancels out

I hope it is clear
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Re: In the figure above, the letters L, M, and N denote the areas of the  [#permalink]

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31 Oct 2018, 03:44
Salsanousi wrote:
Hi shashaankbhat

You know we get Pi * B^2/8 + Pi * A^2/8 correct?

Now if we take Pi/8 as a common factor we have (B^2 + A^2) which equals C^2

So it becomes Pi/8 * C^2/ (Pi/8 * C^2) and this cancels out

I hope it is clear

Ah!! Yes. I missed it. Thank you

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Re: In the figure above, the letters L, M, and N denote the areas of the   [#permalink] 31 Oct 2018, 03:44
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