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In triangle ABC above

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In triangle ABC above  [#permalink]

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New post Updated on: 31 May 2018, 22:24
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28% (02:00) correct 72% (01:21) wrong based on 57 sessions

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In triangle ABC above, if AD=BC ,what is the value of x?
(1) D is the midpoint of side AC.
(2) The angle ABC is a right angle.

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Originally posted by jfdelgado on 31 May 2018, 19:22.
Last edited by amanvermagmat on 31 May 2018, 22:24, edited 1 time in total.
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In triangle ABC above  [#permalink]

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New post Updated on: 01 Jun 2018, 02:27
jfdelgado wrote:
In triangle ABC above, if AD=BC ,what is the value of x?
(1) D is the midpoint of side AC.
(2) The angle ABC is a right angle.


\(ADB\angle=180-2x\) and \(BAD\angle=180-ABD\angle-ADB\angle=x\)
If \(AD=BC\), then \(ACB\angle=BAC\angle=x\).

(1) If \(D\) is the midpoint of side \(AC\), then triangles \(ABD\) and \(BDC\) are identical. So, \(ADB\angle=2x\) and \(x+x+2x=180\). \(x=45\). Sufficient.

(2) If the angle \(ABC\angle\) is a right angle, then \(DBC\angle=90-x\) and \((90-x)+2x+x=180\). \(x=45\). Sufficient.

Answer: D
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Originally posted by Tulkin987 on 31 May 2018, 22:34.
Last edited by Tulkin987 on 01 Jun 2018, 02:27, edited 1 time in total.
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Re: In triangle ABC above  [#permalink]

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New post 31 May 2018, 22:49
Tulkin987 wrote:
jfdelgado wrote:
In triangle ABC above, if AD=BC ,what is the value of x?
(1) D is the midpoint of side AC.
(2) The angle ABC is a right angle.


\(ADB=180-2x\) and \(BAD=180-ABD-ADB=x\)
If \(AD=BC\), then \(ACB=BAC=x\).

(1) If \(D\) is the midpoint of side \(AC\), then triangles \(ABD\) and \(BDC\) are identical. So, \(ADB=2x\) and \(x+x+2x=180\). \(x=45\). Sufficient.

(2) If the angle \(ABC\) is a right angle, then \(DBC=90-x\) and \((90-x)+2x+x=180\). \(x=45\). Sufficient.

Answer: D


Hello

I also agree that answer would be D.
But wouldnt x be equal to 30 degrees from each of the statements. (I am getting 30 degrees)

Can you please check the highlighted part. By the word 'identical', do you mean that both triangles would be congruent? Because a median just divides the triangle into two triangles of equal area, not necessarily identical.
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Re: In triangle ABC above  [#permalink]

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New post 31 May 2018, 22:58
amanvermagmat wrote:
Tulkin987 wrote:
jfdelgado wrote:
In triangle ABC above, if AD=BC ,what is the value of x?
(1) D is the midpoint of side AC.
(2) The angle ABC is a right angle.


\(ADB=180-2x\) and \(BAD=180-ABD-ADB=x\)
If \(AD=BC\), then \(ACB=BAC=x\).

(1) If \(D\) is the midpoint of side \(AC\), then triangles \(ABD\) and \(BDC\) are identical. So, \(ADB=2x\) and \(x+x+2x=180\). \(x=45\). Sufficient.

(2) If the angle \(ABC\) is a right angle, then \(DBC=90-x\) and \((90-x)+2x+x=180\). \(x=45\). Sufficient.

Answer: D


Hello

I also agree that answer would be D.
But wouldnt x be equal to 30 degrees from each of the statements. (I am getting 30 degrees)

Can you please check the highlighted part. By the word 'identical', do you mean that both triangles would be congruent? Because a median just divides the triangle into two triangles of equal area, not necessarily identical.


Hi,

Yes, based on the info given in the first statement triangles \(ABD\) and \(BDC\) are congruent, because \(AB=BC\), \(AD=DC\) and they share \(BD\).

And please check \(x+x+2x=180\). This will result in \(x=45\), but not \(x=30\).

Hope this helps!
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Re: In triangle ABC above  [#permalink]

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New post 31 May 2018, 23:06
Answer: D[/quote]

Hello

I also agree that answer would be D.
But wouldnt x be equal to 30 degrees from each of the statements. (I am getting 30 degrees)

Can you please check the highlighted part. By the word 'identical', do you mean that both triangles would be congruent? Because a median just divides the triangle into two triangles of equal area, not necessarily identical.[/quote]

Hi,

Yes, based on the info given in the first statement triangles \(ABD\) and \(BDC\) are congruent, because \(AB=BC\), \(AD=DC\) and they share \(BD\).

And please check \(x+x+2x=180\). This will result in \(x=45\), but not \(x=30\).

Hope this helps![/quote]

Hello

Actually triangle ABD has its angles measures as x, x and 180-2x (that is because exterior angle BDC is 2x, which is the sum of angle ABD and angle BAD, and since angle ABD is given as x, angle BAD also must be x). And also because of this, AD = BD = BC.

BUT triangle BDC has its angles measures as 2x, 2x and 180-4x (that is because BD=BC so both angles would be 2x each, leaving the third angle DBC to be 180-4x).

Since the angles differ, the triangles cannot be congruent. Thats what I think.
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Re: In triangle ABC above  [#permalink]

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New post 01 Jun 2018, 02:18
Tulkin987 wrote:
jfdelgado wrote:
In triangle ABC above, if AD=BC ,what is the value of x?
(1) D is the midpoint of side AC.
(2) The angle ABC is a right angle.


\(ADB=180-2x\) and \(BAD=180-ABD-ADB=x\)
If \(AD=BC\), then \(ACB=BAC=x\).

(1) If \(D\) is the midpoint of side \(AC\), then triangles \(ABD\) and \(BDC\) are identical. So, \(ADB=2x\) and \(x+x+2x=180\). \(x=45\). Sufficient.

(2) If the angle \(ABC\) is a right angle, then \(DBC=90-x\) and \((90-x)+2x+x=180\). \(x=45\). Sufficient.

Answer: D




I'm not able to understand how ACB=BAC.
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Re: In triangle ABC above  [#permalink]

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New post 01 Jun 2018, 02:22
Chanda15 wrote:
Tulkin987 wrote:
jfdelgado wrote:
In triangle ABC above, if AD=BC ,what is the value of x?
(1) D is the midpoint of side AC.
(2) The angle ABC is a right angle.


\(ADB=180-2x\) and \(BAD=180-ABD-ADB=x\)
If \(AD=BC\), then \(ACB=BAC=x\).

(1) If \(D\) is the midpoint of side \(AC\), then triangles \(ABD\) and \(BDC\) are identical. So, \(ADB=2x\) and \(x+x+2x=180\). \(x=45\). Sufficient.

(2) If the angle \(ABC\) is a right angle, then \(DBC=90-x\) and \((90-x)+2x+x=180\). \(x=45\). Sufficient.

Answer: D




I'm not able to understand how ACB=BAC.


ACB and BAC are angles. Will correct.
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Re: In triangle ABC above  [#permalink]

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New post 01 Jun 2018, 06:38
According to statement 1, triangle BDC becomes equilateral triangle, and x = 30 not 45. If two sides are equal, doesn't mean the third angle BDA also becomes 2x.

Statement 2 is also sufficient.

Hence, D is the answer.

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Re: In triangle ABC above  [#permalink]

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New post 30 Jun 2018, 11:07
amanvermagmat

Could you please elaborate why D is answer
If possible provide solution

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Re: In triangle ABC above &nbs [#permalink] 30 Jun 2018, 11:07
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