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What is the area of triangle ABC ?

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What is the area of triangle ABC ?  [#permalink]

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New post Updated on: 23 Jun 2015, 23:07
3
00:00
A
B
C
D
E

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  65% (hard)

Question Stats:

57% (01:51) correct 43% (01:54) wrong based on 170 sessions

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What is the area of triangle ABC ?

(1) Side DC = 20

(2) Side AC = 8

Hi,

In the explanation of this question, it's telling me that the two angles at C are the same and therefore, the triangles are similar. But i'm not understanding why, how do we know thse two angles are equal?

Please help, thank you very much.

Originally posted by yvette726 on 21 Feb 2014, 08:12.
Last edited by Bunuel on 23 Jun 2015, 23:07, edited 2 times in total.
Renamed the topic and edited the question.
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Re: What is the area of triangle ABC ?  [#permalink]

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New post 21 Feb 2014, 09:08
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4
Image
What is the area of triangle ABC ?

Angle ACE is a straight line, hence it's 180°. Now, since (angle BCD)=90°, then (angle ACB) + (angle DCE) = 180° - 90° = 90°.

Next, in triangle ABC, (angle ACB) + (angle ABC) = 90°. Thus we have that:
(angle ACB) + (angle ABC) = 90° = (angle ACB) + (angle DCE) --> (angle ABC) = (angle DCE), which on the other hand implies that (angle ACB) = (angle CDE).

Therefore, all three angles in triangles ABC and CDE are equal, so these triangles are similar.

If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\).
OR in another way: in two similar triangles, the ratio of their areas is the square of the ratio of their sides: \(\frac{AREA}{area}=\frac{SIDE^2}{side^2}\).

(1) Side DC = 20 --> \(CE=\sqrt{20^2-16^2}=12\) --> \(AC=20-12=8\) --> \(\frac{AREA_{CDE}}{area_{ABC}}=\frac{16^2}{8^2}\). Since the area of triangle CDE = 16*12/2 = 96, then \(\frac{96}{area_{ABC}}=\frac{16^2}{8^2}\) --> we can find the area od triangle ABC. Sufficient.

(2) Side AC = 8. The same info as above. Sufficient.

Answer: D.

Below image might help to understand better:
Attachment:
Untitled.png
Untitled.png [ 24.88 KiB | Viewed 5573 times ]

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Re: What is the area of triangle ABC ?  [#permalink]

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New post 21 Feb 2014, 09:35
Thank you Bunuel ! Superb explanation.
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Re: What is the area of triangle ABC ?  [#permalink]

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New post 31 Mar 2016, 19:54
Bunuel wrote:
Image
What is the area of triangle ABC ?

Angle ACE is a straight line, hence it's 180°. Now, since (angle BCD)=90°, then (angle ACB) + (angle DCE) = 180° - 90° = 90°.

Next, in triangle ABC, (angle ACB) + (angle ABC) = 90°. Thus we have that:
(angle ACB) + (angle ABC) = 90° = (angle ACB) + (angle DCE) --> (angle ABC) = (angle DCE), which on the other hand implies that (angle ACB) = (angle CDE).

Therefore, all three angles in triangles ABC and CDE are equal, so these triangles are similar.

If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\).
OR in another way: in two similar triangles, the ratio of their areas is the square of the ratio of their sides: \(\frac{AREA}{area}=\frac{SIDE^2}{side^2}\).

(1) Side DC = 20 --> \(CE=\sqrt{20^2-16^2}=12\) --> \(AC=20-12=8\) --> \(\frac{AREA_{CDE}}{area_{ABC}}=\frac{16^2}{8^2}\). Since the area of triangle CDE = 16*12/2 = 96, then \(\frac{96}{area_{ABC}}=\frac{16^2}{8^2}\) --> we can find the area od triangle ABC. Sufficient.

(2) Side AC = 8. The same info as above. Sufficient.

Answer: D.

Below image might help to understand better:
Attachment:
Untitled.png


Hey I have a question here you mentioned ABC is similar to CDE
But i think the rotation specified here is incorrect
i think ABC is similar to ECD
Am i understanding this correctly ?
Would really appreciate you being more responsive
regards
S.C.S.A
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Re: What is the area of triangle ABC ?  [#permalink]

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New post 31 Mar 2016, 20:26
1
Chiragjordan wrote:
Bunuel wrote:
Image
What is the area of triangle ABC ?

Angle ACE is a straight line, hence it's 180°. Now, since (angle BCD)=90°, then (angle ACB) + (angle DCE) = 180° - 90° = 90°.

Next, in triangle ABC, (angle ACB) + (angle ABC) = 90°. Thus we have that:
(angle ACB) + (angle ABC) = 90° = (angle ACB) + (angle DCE) --> (angle ABC) = (angle DCE), which on the other hand implies that (angle ACB) = (angle CDE).

Therefore, all three angles in triangles ABC and CDE are equal, so these triangles are similar.

If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\).
OR in another way: in two similar triangles, the ratio of their areas is the square of the ratio of their sides: \(\frac{AREA}{area}=\frac{SIDE^2}{side^2}\).

(1) Side DC = 20 --> \(CE=\sqrt{20^2-16^2}=12\) --> \(AC=20-12=8\) --> \(\frac{AREA_{CDE}}{area_{ABC}}=\frac{16^2}{8^2}\). Since the area of triangle CDE = 16*12/2 = 96, then \(\frac{96}{area_{ABC}}=\frac{16^2}{8^2}\) --> we can find the area od triangle ABC. Sufficient.

(2) Side AC = 8. The same info as above. Sufficient.

Answer: D.

Below image might help to understand better:
Attachment:
Untitled.png


Hey I have a question here you mentioned ABC is similar to CDE
But i think the rotation specified here is incorrect
i think ABC is similar to ECD
Am i understanding this correctly ?
Would really appreciate you being more responsive
regards
S.C.S.A


Hi,

If you see the highlighted portion, Bunuel writes these triangles, no notation is given, are similar ...
Yes, it is always better to write the rotation of triangle as per the similarity of angles..
In this particular case, triangle ABC is similar to triangle ECD...

But finally we should look at the question and decide which angles are equal
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Re: What is the area of triangle ABC ?  [#permalink]

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New post 08 Sep 2017, 18:22
yvette726 wrote:
Attachment:
Untitled.png
What is the area of triangle ABC ?

(1) Side DC = 20

(2) Side AC = 8

Hi,

In the explanation of this question, it's telling me that the two angles at C are the same and therefore, the triangles are similar. But i'm not understanding why, how do we know thse two angles are equal?

Please help, thank you very much.


Bunuel is it safe to say as a universal truth, on the GMAT, that if you know the hypotenuse of a right triangle AND the length of one other side then you know the length of the other side? I feel obviously yes but just want to make sure.
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Re: What is the area of triangle ABC ?  [#permalink]

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New post 08 Sep 2017, 20:08
Nunuboy1994 wrote:
yvette726 wrote:
Attachment:
Untitled.png
What is the area of triangle ABC ?

(1) Side DC = 20

(2) Side AC = 8

Hi,

In the explanation of this question, it's telling me that the two angles at C are the same and therefore, the triangles are similar. But i'm not understanding why, how do we know thse two angles are equal?

Please help, thank you very much.


Bunuel is it safe to say as a universal truth, on the GMAT, that if you know the hypotenuse of a right triangle AND the length of one other side then you know the length of the other side? I feel obviously yes but just want to make sure.



Hi..

Yes, if you know any two sides in a right angle triangle, you can find the third side.
Also if you are given two sides of any triangle with the angle in between them, you can find the third side
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Re: What is the area of triangle ABC ?   [#permalink] 08 Sep 2017, 20:08
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