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# If side BX has length 10 and side AC has length 8, what is the area of

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

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09 Sep 2018, 08:09
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Difficulty:

65% (hard)

Question Stats:

65% (02:13) correct 35% (03:10) wrong based on 71 sessions

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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. $$12\sqrt{3}$$

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Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 857
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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09 Sep 2018, 09:06
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1
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
Joined: 19 Sep 2016
Posts: 17
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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15 Sep 2018, 07:49
Please explain the solution. Thanks. !!
Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 857
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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15 Sep 2018, 08:51
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

Rise above the storm, you will find the sunshine

Director
Joined: 04 Sep 2015
Posts: 522
Location: India
WE: Information Technology (Computer Software)
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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15 Sep 2018, 12:09
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
Status: Learning stage
Joined: 01 Oct 2017
Posts: 857
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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15 Sep 2018, 13:12
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.
_________________

Regards,

PKN

Rise above the storm, you will find the sunshine

Director
Joined: 04 Sep 2015
Posts: 522
Location: India
WE: Information Technology (Computer Software)
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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15 Sep 2018, 13:29
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
Status: Learning stage
Joined: 01 Oct 2017
Posts: 857
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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15 Sep 2018, 13:32
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

Manager
Joined: 27 Dec 2016
Posts: 222
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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15 Sep 2018, 19:08
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
Status: Learning stage
Joined: 01 Oct 2017
Posts: 857
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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17 Sep 2018, 23:08
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.
_________________

Regards,

PKN

Rise above the storm, you will find the sunshine

Manager
Joined: 27 Dec 2016
Posts: 222
Re: If side BX has length 10 and side AC has length 8, what is the area of  [#permalink]

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18 Sep 2018, 07:29
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
Status: Learning stage
Joined: 01 Oct 2017
Posts: 857
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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18 Sep 2018, 18:36
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.
_________________

Regards,

PKN

Rise above the storm, you will find the sunshine

Re: If side BX has length 10 and side AC has length 8, what is the area of &nbs [#permalink] 18 Sep 2018, 18:36
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