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Math Expert V
Joined: 02 Sep 2009
Posts: 60560
If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 60% (03:10) correct 40% (03:01) wrong based on 91 sessions

### HideShow timer Statistics If side BX has length 10 and side AC has length 8, what is the area of the shaded region?

A. $$2\sqrt{3}$$

B. $$4\sqrt{3}$$

C. $$6\sqrt{3}$$

D. $$8\sqrt{3}$$

E. $$12\sqrt{3}$$

Attachment: image019.jpg [ 4.04 KiB | Viewed 1495 times ]

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Director  D
Status: Learning stage
Joined: 01 Oct 2017
Posts: 987
WE: Supply Chain Management (Energy and Utilities)
If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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Bunuel wrote: If side BX has length 10 and side AC has length 8, what is the area of the shaded region?

A. $$2\sqrt{3}$$

B. $$4\sqrt{3}$$

C. $$6\sqrt{3}$$

D. $$8\sqrt{3}$$

E. $$12\sqrt{3}$$

Area of the shaded region=Area of right angled triangle BOC (30-90-60) (Refer enclosed figure)

ABC is a 30-90-60 triangle, Hence $$BC=\frac{AC}{2}=\frac{8}{2}=4$$

Now in the triangle BOC(30-90-60) , Base=OC=$$\frac{4}{2}$$=2, height=OB=$$2\sqrt{3}$$
So, area of triangle BOC(shaded region)=$$\frac{1}{2} *OC*OB=\frac{1}{2} *2*2\sqrt{3}$$=$$2\sqrt{3}$$

Ans. (A)
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PKN

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Intern  B
Joined: 19 Sep 2016
Posts: 46
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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Please explain the solution. Thanks. !!
Director  D
Status: Learning stage
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WE: Supply Chain Management (Energy and Utilities)
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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siddharthfrancis wrote:
Please explain the solution. Thanks. !!

Hi siddharthfrancis,
Detailed explanation has been provided in the above post. You may raise specific queries(if any).
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PKN

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Director  P
Joined: 04 Sep 2015
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Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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PKN wrote:
siddharthfrancis wrote:
Please explain the solution. Thanks. !!

Hi siddharthfrancis,
Detailed explanation has been provided in the above post. You may raise specific queries(if any).

trianlge 30-60-90 is fine, how did we get BC=AC/2?

what rule?
Director  D
Status: Learning stage
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WE: Supply Chain Management (Energy and Utilities)
If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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1
PKN wrote:
siddharthfrancis wrote:
Please explain the solution. Thanks. !!

Hi siddharthfrancis,
Detailed explanation has been provided in the above post. You may raise specific queries(if any).

trianlge 30-60-90 is fine, how did we get BC=AC/2?

what rule?

In the triangle ABC , Angle A=30, Angle C=60, Angle B=90
So the corresponding sides are in the ratio:
BC:AB:AC=x:√3x:2x
Or, BC=x and AC=2x
Or, AC is 2 times BC
Or, BC=AC/2

You may go thru the below link for a detailed insight on SPECIAL TRIANGLES.
https://gmatclub.com/forum/math-triangles-87197.html

Thank you.
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PKN

Rise above the storm, you will find the sunshine
Director  P
Joined: 04 Sep 2015
Posts: 675
Location: India
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Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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In the triangle ABC , Angle A=30, Angle C=60, Angle B=90
So the corresponding sides are in the ratio:
BC:AB:AC=x:√3x:2x
Or, BC=x and AC=2x
Or, AC is 2 times BC
Or, BC=AC/2

You may go thru the below link for a detailed insight on SPECIAL TRIANGLES.
https://gmatclub.com/forum/math-triangles-87197.html

Thank you.[/quote]

Now i understand that we use the 30-60-90 rule on the bigger triangle to find one comman side, and then wth that side we use 30-60-90 on the smaller trianlge to find the lenghts of the sides of the triangle and thus finding the area of the smaller triangle.

thanks.
Director  D
Status: Learning stage
Joined: 01 Oct 2017
Posts: 987
WE: Supply Chain Management (Energy and Utilities)
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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1
In the triangle ABC , Angle A=30, Angle C=60, Angle B=90
So the corresponding sides are in the ratio:
BC:AB:AC=x:√3x:2x
Or, BC=x and AC=2x
Or, AC is 2 times BC
Or, BC=AC/2

You may go thru the below link for a detailed insight on SPECIAL TRIANGLES.
https://gmatclub.com/forum/math-triangles-87197.html

Thank you.

Now i understand that we use the 30-60-90 rule on the bigger triangle to find one comman side, and then wth that side we use 30-60-90 on the smaller trianlge to find the lenghts of the sides of the triangle and thus finding the area of the smaller triangle.

thanks.[/quote]

You are welcome.
_________________
Regards,

PKN

Rise above the storm, you will find the sunshine
Senior Manager  P
Joined: 27 Dec 2016
Posts: 304
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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Hi Bunuel

Could you please explain where I made a mistake in my solution? I got B as my answer. My strategy was to find the area of ABC and BXY. Then subtracting these two to find the area of BOC.

Area of ABC= 1/2*(4√3)*4 = 8√3
Area of BXY = 1/2*(5√3)*5= 12.5√3

Area of the shaded region: 12.5√3 - 8√3= 4.5√3

Thank You!
Director  D
Status: Learning stage
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Posts: 987
WE: Supply Chain Management (Energy and Utilities)
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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csaluja wrote:
Hi Bunuel

Could you please explain where I made a mistake in my solution? I got B as my answer. My strategy was to find the area of ABC and BXY. Then subtracting these two to find the area of BOC.
Area of ABC= 1/2*(4√3)*4 = 8√3
Area of BXY = 1/2*(5√3)*5= 12.5√3

Area of the shaded region: 12.5√3 - 8√3= 4.5√3

Thank You!

Hi csaluja ,
I have highlighted the mistake in reasoning in red color.
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PKN

Rise above the storm, you will find the sunshine
Senior Manager  P
Joined: 27 Dec 2016
Posts: 304
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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PKN wrote:
csaluja wrote:
Hi Bunuel

Could you please explain where I made a mistake in my solution? I got B as my answer. My strategy was to find the area of ABC and BXY. Then subtracting these two to find the area of BOC.
Area of ABC= 1/2*(4√3)*4 = 8√3
Area of BXY = 1/2*(5√3)*5= 12.5√3

Area of the shaded region: 12.5√3 - 8√3= 4.5√3

Thank You!

Hi csaluja ,
I have highlighted the mistake in reasoning in red color.

Hi PKN,

Could you please explain why the highlighted part is incorrect? Would greatly appreciate it!
Director  D
Status: Learning stage
Joined: 01 Oct 2017
Posts: 987
WE: Supply Chain Management (Energy and Utilities)
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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csaluja wrote:
PKN wrote:
csaluja wrote:
Hi Bunuel

Could you please explain where I made a mistake in my solution? I got B as my answer. My strategy was to find the area of ABC and BXY. Then subtracting these two to find the area of BOC.
Area of ABC= 1/2*(4√3)*4 = 8√3
Area of BXY = 1/2*(5√3)*5= 12.5√3

Area of the shaded region: 12.5√3 - 8√3= 4.5√3

Thank You!

Hi csaluja ,
I have highlighted the mistake in reasoning in red color.

Hi PKN,

Could you please explain why the highlighted part is incorrect? Would greatly appreciate it!

Hi csaluja,

Area of BOC= Area of ABC-Area of BXY

Let's break the area of ABC and BXY in terms of area of BOC,

Area of ABC= Area of AOB+Area of BOC--------(1)
Area of BXY=Area of XOCY+Area of BOC--------(2)

Now, subtracting (2) from(1) ,we have
Area of ABC-Area of BXY=Area of AOB-Area of XOCY, which is not equal to area of BOC.

Hope it's clear.
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PKN

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Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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