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# In the diagram below, triangle ABC is right-angled at B. What is the

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Math Expert
Joined: 02 Sep 2009
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In the diagram below, triangle ABC is right-angled at B. What is the  [#permalink]

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17 Jul 2017, 23:31
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In the diagram below, triangle ABC is right-angled at B. What is the measure of ∠BAC?

(1) BD = BC and AE = AD.
(2) BE = DE.

Attachment:

2017-07-18_1030.png [ 6.99 KiB | Viewed 2080 times ]

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Re: In the diagram below, triangle ABC is right-angled at B. What is the  [#permalink]

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18 Jul 2017, 05:03
i have tried every possible thing to get the answer but knowing that two sides are same, i cant get the angle of the other side. I cant find the relationship between the two small triangles even...so E for me
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In the diagram below, triangle ABC is right-angled at B. What is the  [#permalink]

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18 Jul 2017, 05:36
1
Bunuel wrote:
In the diagram below, triangle ABC is right-angled at B. What is the measure of ∠BAC?

(1) BD = BC and AE = AD.
(2) BE = DE.

Attachment:
The attachment 2017-07-18_1030.png is no longer available

It should be C

Statement 1: Insufficient
(1) BD = BC and AE = AD.

This tells us both top and bottom triangles are isosceles but it does not provide us any information about the central triangle.

Statement 2: Insufficient
This tells us central triangle is an isosceles but it does not give out information about the angles.

Lets combined 1 and 2.

From 1, Since $$BD = BC$$, we know $$∠BDC = ∠BCD$$. Let us call this angle x. In the triangle $$BDC$$, third angle $$∠CBD$$ would be equal to $$180 - 2x$$.

Since triangle ABC is right angled at B, $$∠EBD = 90 - (180 - 2x) = -90 + 2x$$

From 2, we know $$BE = DE$$, hence $$∠EBD = ∠EDB = -90 + 2x$$

The third angle in the central triangle would be $$180 - 2 (-90 + 2x) = 360 - 4x$$

Lets name the two equal angles from triangle $$AED = y$$

At point D, we have

$$x + (-90 + 2x) + y = 180$$
$$3x + y = 270$$ --- (1)

At point E, we have

$$360 - 4x + y = 180$$
$$4x - y = 180$$ --- (2)

Solving the two equations, we get the value of y and since two angles in triangle AED are y, the third $$∠BAC$$ can be calculated using $$180 - 2y$$.

Edit: resizing diagram as it was too large.
Attachments

diagram.jpg [ 28.2 KiB | Viewed 1727 times ]

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Re: In the diagram below, triangle ABC is right-angled at B. What is the  [#permalink]

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18 Jul 2017, 22:06
anybody can solve this a simpler way?
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Posts: 7958
In the diagram below, triangle ABC is right-angled at B. What is the  [#permalink]

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18 Jul 2017, 23:59
Bunuel wrote:
In the diagram below, triangle ABC is right-angled at B. What is the measure of ∠BAC?

(1) BD = BC and AE = AD.
(2) BE = DE.

Hi...
A simpler way as someone has asked
COMBINED..
Angle DBC=angle DCB=x
EBD=EDB=r
Now the equations..
1) from angle ABC=90..
r+x=90.....(i)
2) from triangle AED and ∆EBD
y=2R.....(ii)
r+y=2x.....(iii)

Three variables x,y and r.... Three equations above..
Sufficient

C
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Re: In the diagram below, triangle ABC is right-angled at B. What is the  [#permalink]

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19 Jul 2017, 00:08
Yes it should be C...Bunuel You are giving Good questions at least your questions are building my concepts....i am able to apply all the rules that i have learned so far...Thanks
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Re: In the diagram below, triangle ABC is right-angled at B. What is the  [#permalink]

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19 Jul 2017, 00:38
mohshu wrote:
anybody can solve this a simpler way?

Yes, instead of calculating Y, you can calculate X and since we know angle B in the larger triangle ABC is 90, knowing X, we can simply calculate required ∠BAC by 180 - 90 - x. This would not require you to calculate Y.
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Re: In the diagram below, triangle ABC is right-angled at B. What is the  [#permalink]

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27 Feb 2019, 13:54
chetan2u confused by your explanation, see highlighted below

chetan2u wrote:
Hi...
A simpler way as someone has asked
COMBINED..
Angle DBC=angle DCB=x shouldn't it be ∠BDC = ∠DCB? since prompt is DB=BC
EBD=EDB=r
Now the equations..
1) from angle ABC=90..
r+x=90.....(i)
2) from triangle AED and ∆EBD
y=2R.....(ii)
r+y=2x.....(iii)

Three variables x,y and r.... Three equations above..
Sufficient

C
Re: In the diagram below, triangle ABC is right-angled at B. What is the   [#permalink] 27 Feb 2019, 13:54
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