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In the figure, point D divides side BC of triangle ABC into segments

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Re: In the figure, point D divides side BC of triangle ABC into segments  [#permalink]

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New post 12 Nov 2016, 20:57
enigma123 wrote:
Image
In the figure, point D divides side BC of triangle ABC into segments BD and DC of lengths 1 and 2 units respectively. Given that ÐADC = 60º and ÐABD = 45º, what is the measure of angle x in degrees? (Note:
Figure is not drawn to scale.)

A. 55
B. 60
C. 70
D. 75
E. 90

Any idea guys what will be the correct answer please? Also any idea how can I cut and paste the pictures in my post? Is it possible?

Attachment:
The attachment Triangle.png is no longer available


Answer: Option D

Please check solution
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Re: In the figure, point D divides side BC of triangle ABC into segments  [#permalink]

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New post 17 Aug 2017, 09:30
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Attachment:
IMG_6850.JPG
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. see attached pic

Answer is 75 degrees.
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Re: In the figure, point D divides side BC of triangle ABC into segments  [#permalink]

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New post 06 Dec 2017, 20:10
1
Hi bunuel,

Sorry for bumping. A quick question.

How do we know that AD is not equal to 1. If it is equal to 1 then the ratio 2:1 when a perpendicular is drawn does not hold good right? Am i missing something?
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Re: In the figure, point D divides side BC of triangle ABC into segments  [#permalink]

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New post 06 Dec 2017, 20:40
gauthamvm wrote:
Hi bunuel,

Sorry for bumping. A quick question.

How do we know that AD is not equal to 1. If it is equal to 1 then the ratio 2:1 when a perpendicular is drawn does not hold good right? Am i missing something?


Image

x=45+30=75.

Notes:
Sides with one blue segment crossing them are equal and sides with two blues segments crossing them are equal too.

CO is perpendicular to AD --> OD=1 (from 30-60-90 right triangle property as sides are in ratio \(1:\sqrt{3}:2\)) --> as OD=BD=1 then ODB is an isosceles triangle.
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Re: In the figure, point D divides side BC of triangle ABC into segments  [#permalink]

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New post 17 Apr 2018, 06:44
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enigma123 wrote:
Image

In the figure, point D divides side BC of triangle ABC into segments BD and DC of lengths 1 and 2 units respectively. Given that ÐADC = 60º and ÐABD = 45º, what is the measure of angle x in degrees? (Note: Figure is not drawn to scale.)

A. 55
B. 60
C. 70
D. 75
E. 90


Attachment:
Triangle.png

Attachment:
task%236.png


First let's add a line from point C so that it is PERPENDICULAR to line AD
Also, note that, ∠EDB is 120° since it lies on a line with a 60° angle.
Image


Since we already know 2 angles in ∆CED, we can see that the remaining angle, ∠ECD, is 30°, which means ∆CED is a 30-60-90 right triangle.
So, ∆CED is a 30-60-90 right triangle AND we know that the hypotenuse CD has length 2, we can conclude that side ED has length 1.
Image


Now draw a line from E to B.
Since ED = EB (both have length 1), we can see that ∆EDB is an ISOSCELES TRIANGLE, which means the two remaining angles are each 30°
Image


Next, ∠EAB is 150° since it shares a line with a 30° angle
Also, since we were originally told that ∠CBA is 45°, we can conclude that ∠ABE is 15°
Image

Now focus on ∆CBE
Notice that this triangle is an ISOSCELES triangle, because ∠ECB = ∠EBC = 30°
This means that side EC = side EB
Image


Now focus on ∆EAB
Since we already know two of the angles in this triangle (150° and 15°), we can conclude that = ∠EAB = 15°
Image


Stay focused on ∆EAB
This triangle is an ISOSCELES triangle, because ∠EAB = ∠EBA = 15°
This means that side AE = side EB
Image


Now focus on ∆ACE
Since CE = EA, this is an ISOSCELES triangle
Since one angle = 90°, the other two angles are each 45°
Image

At this point, we can see that x = 45° + 30° = 75°
Answer: D

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Re: In the figure, point D divides side BC of triangle ABC into segments  [#permalink]

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New post 29 May 2018, 02:49
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vigrah wrote:
say angle CAB=y
since sum of angles in a triangle is 180
x+y+45=180
x+y=135 equation 1

line AD is dividing BC in 2:1 ratio
hence
X+2/3Y+60=180
X+2/3y=120 equation 2
solving equation 1 &2 we get x=75
answer is D


This answer is wrong. Solving this system does not yield x=75, but x=90, and y=45.
This answer should not have so many kudos, it is very misleading.
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Re: In the figure, point D divides side BC of triangle ABC into segments  [#permalink]

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New post 20 Aug 2019, 21:03
vigrah wrote:
say angle CAB=y
since sum of angles in a triangle is 180
x+y+45=180
x+y=135 equation 1

line AD is dividing BC in 2:1 ratio
hence
X+2/3Y+60=180
X+2/3y=120 equation 2
solving equation 1 &2 we get x=75
answer is D


Bunnel/Experts,

I have a doubt here.I understand angles and sides ratio will be proportional.

So If I see angle opposite to side with lenght 1 is 15 degree .So why not we
directly comprehend angle opposite to CB(3 units) is 45 degrees.

from this I will get angle y as 45 degrees
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Re: In the figure, point D divides side BC of triangle ABC into segments  [#permalink]

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New post 20 Aug 2019, 22:00
prabsahi wrote:
vigrah wrote:
say angle CAB=y
since sum of angles in a triangle is 180
x+y+45=180
x+y=135 equation 1

line AD is dividing BC in 2:1 ratio
hence
X+2/3Y+60=180
X+2/3y=120 equation 2
solving equation 1 &2 we get x=75
answer is D


Bunnel/Experts,

I have a doubt here.I understand angles and sides ratio will be proportional.

So If I see angle opposite to side with lenght 1 is 15 degree .So why not we
directly comprehend angle opposite to CB(3 units) is 45 degrees.

from this I will get angle y as 45 degrees


The solution you are quoting is not correct. Please check the discussion on previous pages.
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In the figure, point D divides side BC of triangle ABC into segments  [#permalink]

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New post 13 Oct 2019, 09:33
Bunuel wrote:
enigma123 wrote:
Image

In the figure, point D divides side BC of triangle ABC into segments BD and DC of lengths 1 and 2 units respectively. Given that ÐADC = 60º and ÐABD = 45º, what is the measure of angle x in degrees? (Note:
Figure is not drawn to scale.)

(A) 55
(B) 60
(C) 70
(D) 75
(E) 90

Any idea guys what will be the correct answer please? Also any idea how can I cut and paste the pictures in my post? Is it possible?


Complete solution for all the angles is in the image below:

Image

x=45+30=75.

Notes:
Sides with one blue segment crossing them are equal and sides with two blues segments crossing them are equal too.

CO is perpendicular to AD --> OD=1 (from 30-60-90 right triangle property as sides are in ratio \(1:\sqrt{3}:2\)) --> as OD=BD=1 then ODB is an isosceles triangle.

<CDO and <BDO are supplementary to each other (supplementary angles are two angles that add up to 180°), so <BDO=120 --> <DAB=180-(120+45)=15.

As ODB is an isosceles triangle --> <DOB=<DBO=30. <OBA=45-30=15 --> AOB is an isosceles triangle, so OA=OB. Also COB is an isosceles triangle, so CO=OB --> OA=CO=OB. So, AOC is an isosceles triangle --> <CAO=<OCA=45 (as <COA=90) --> x=45+30=75.

Answer: D.

P.S. You can attach image files directly to the post.

Attachment:
Triangle complete.PNG


Hi Bunuel,

I have two questions which came in when I was solving the problem. Tried reading through the other explanations but am still unclear.

1. When you drew the segment OB, how did you determine that it will split angle B (i.e. 45) into 30 and 15? Why could it not be any other split like 20 and 25 or 10 and 35 etc?
2. In triangle AOC we know that angle O is 90, but then how did you conclude that the other 2 angles are 45 each and not any other split?

Warm Regards,
Pritishd
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In the figure, point D divides side BC of triangle ABC into segments   [#permalink] 13 Oct 2019, 09:33

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