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what is the area of the triangle ABC?

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

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New post 07 Mar 2018, 08:20
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What is the area of the triangle ABC?

(1) The sides AB and BC are in ratio 1:2 while the sum of sides AB and BC is 60.
(2) The angle BAC = 120 deg.

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

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New post 07 Mar 2018, 09:17
3
When you combine the info, there is only ONE possible triangle...
AB =20 and BC =40...
Draw a line with one end as A..
Make a 120° angle from A and cut 20 units on that line as B..
From B take 40 units on a line, it will cut the line starting with A at one end at a point C..
Since you have a unique triangle, you can find Area..
Sufficient

C

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

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New post 07 Mar 2018, 09:30
gmatbusters wrote:
What is the area of the triangle ABC?

(1) The sides AB and BC are in ratio 1:2 while the sum of sides AB and BC is 60.
(2) The angle BAC = 120 deg.


Imo C

From statement 1 we can not calculate area as we are not given the third side .
We can calculate the individual values of the two sides but that is unhelpful as we need third side for area using heron's formula.

From 2
We know the measure of every angle but we dont know the sides so insufficient

Together they are sufficient as we know two sides and it is and isosceles triangle so we we draw perpendicular from A and calculate the altitude using 30-60-90 triangle
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Re: what is the area of the triangle ABC?  [#permalink]

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New post 07 Mar 2018, 09:54
1
How you got it isosceles?

arvind910619 wrote:
gmatbusters wrote:
What is the area of the triangle ABC?

(1) The sides AB and BC are in ratio 1:2 while the sum of sides AB and BC is 60.
(2) The angle BAC = 120 deg.


Imo C

From statement 1 we can not calculate area as we are not given the third side .
We can calculate the individual values of the two sides but that is unhelpful as we need third side for area using heron's formula.

From 2
We know the measure of every angle but we dont know the sides so insufficient

Together they are sufficient as we know two sides and it is and isosceles triangle so we we draw perpendicular from A and calculate the altitude using 30-60-90 triangle

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

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New post 09 Mar 2018, 08:12
Standard formula : area of any triangle = (1/2) a*b* Sin(included angle (a,b)).
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what is the area of the triangle ABC?  [#permalink]

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New post Updated on: 09 Mar 2018, 08:31
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Hi Guys

Whenever any DS question asks about Area, Perimeter or any other details related to a triangle, we should visualize if the triangle can be uniquely drawn or not.


This Question is based on a very NOTE IMPORTANT EXCEPTION:

The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or
Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the
known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In
all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known
as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the
third side and fall back on SSS.

If any triangle follows laws of Congruency, it means it can be drawn Uniquely.



Happy Learning
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Originally posted by gmatbusters on 09 Mar 2018, 08:28.
Last edited by gmatbusters on 09 Mar 2018, 08:31, edited 1 time in total.
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Re: what is the area of the triangle ABC?  [#permalink]

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New post 09 Mar 2018, 08:29
please read the second statement carefully, the angle given is not the included angle.

srinjoy1990 wrote:
Standard formula : area of any triangle = (1/2) a*b* Sin(included angle (a,b)).

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

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New post 09 Mar 2018, 08:46
Now the Question arises, why the SSA congruency rule is valid for obtuse and right angled triangle, but not for acute-angled triangle.
It can be easily seen through the attached sketch, how the right angle and obtuse angle define the triangle clearly. :cool:


gmatbusters wrote:
Hi Guys

Whenever any DS question asks about Area, Perimeter or any other details related to a triangle, we should visualize if the triangle can be uniquely drawn or not.


This Question is based on a very NOTE IMPORTANT EXCEPTION:

The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or
Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the
known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In
all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known
as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the
third side and fall back on SSS.

If any triangle follows laws of Congruency, it means it can be drawn Uniquely.



Happy Learning

Attachments

WhatsApp Image 2018-03-09 at 21.13.07.jpeg
WhatsApp Image 2018-03-09 at 21.13.07.jpeg [ 109.55 KiB | Viewed 877 times ]


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

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New post 12 Mar 2018, 13:22
gmatbusters wrote:
What is the area of the triangle ABC?

(1) The sides AB and BC are in ratio 1:2 while the sum of sides AB and BC is 60.
(2) The angle BAC = 120 deg.


Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

We suppose a triangle has 3 variables, since we can determine a triangle, if we know 3 lengths of the triangle's sides.
Since we have 2 equations from the condition 1 which are \(AB:BC = 1:2\) and \(AB + BC = 60\) and 1 equation from the condition 2), both conditions together 1) & 2) are sufficient.

We have \(AB = 20\) and \(BC = 40\) from the condition 1)

The area of the triangle is \(\frac{1}{2} \cdot AB \cdot BC \cdot sin(120°) = \frac{1}{4} \cdot 20 \cdot 40 \cdot \sqrt{3} = 200\sqrt{3}\)

Therefore, the answer is C.
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Re: what is the area of the triangle ABC?  [#permalink]

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New post 13 Mar 2018, 06:10
Hii
I am afraid, your approach doesn't seem right.

In the formula, A = (Side1 * Side2 * Sin Angle)/2, the angle should be the included angle.



MathRevolution wrote:
gmatbusters wrote:
What is the area of the triangle ABC?

(1) The sides AB and BC are in ratio 1:2 while the sum of sides AB and BC is 60.
(2) The angle BAC = 120 deg.


Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

We suppose a triangle has 3 variables, since we can determine a triangle, if we know 3 lengths of the triangle's sides.
Since we have 2 equations from the condition 1 which are \(AB:BC = 1:2\) and \(AB + BC = 60\) and 1 equation from the condition 2), both conditions together 1) & 2) are sufficient.

We have \(AB = 20\) and \(BC = 40\) from the condition 1)

The area of the triangle is \(\frac{1}{2} \cdot AB \cdot BC \cdot sin(120°) = \frac{1}{4} \cdot 20 \cdot 40 \cdot \sqrt{3} = 200\sqrt{3}\)

Therefore, the answer is C.

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

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New post 13 Mar 2018, 06:33
pushpitkc niks18 Hatakekakashi
amanvermagmat Bunuel

Any way to link opposite sides with angles here since ratio between sides is given?

If AB = x, BC = 2x and AB + BC = 60. We can get AB = 20 and BC = 40.
How do I get third side length here?
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what is the area of the triangle ABC?  [#permalink]

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New post 13 Mar 2018, 07:14
adkikani

Please see the explanation below, we can very easily find the third side by constructing the triangle.
BINGO!!!, actually u need not draw the triangle to the scale, but yes u can see that you can draw, So the triangle can be uniquely drawn.
Hence all parameter related to triangle can be found out.

You might also like this question: https://gmatclub.com/forum/what-is-the- ... l#p2027579

Happy Learning!!!

gmatbusters wrote:
Now the Question arises, why the SSA congruency rule is valid for obtuse and right angled triangle, but not for acute-angled triangle.
It can be easily seen through the attached sketch, how the right angle and obtuse angle define the triangle clearly. :cool:


gmatbusters wrote:
Hi Guys

Whenever any DS question asks about Area, Perimeter or any other details related to a triangle, we should visualize if the triangle can be uniquely drawn or not.


This Question is based on a very NOTE IMPORTANT EXCEPTION:

The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or
Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the
known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In
all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known
as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the
third side and fall back on SSS.

If any triangle follows laws of Congruency, it means it can be drawn Uniquely.



Happy Learning

Attachments

WhatsApp Image 2018-03-09 at 21.13.07 (1).jpeg
WhatsApp Image 2018-03-09 at 21.13.07 (1).jpeg [ 109.55 KiB | Viewed 656 times ]


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

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New post 13 Mar 2018, 12:25
4
adkikani wrote:
pushpitkc niks18 Hatakekakashi
amanvermagmat Bunuel

Any way to link opposite sides with angles here since ratio between sides is given?

If AB = x, BC = 2x and AB + BC = 60. We can get AB = 20 and BC = 40.
How do I get third side length here?


Hi adkikani

In GMAT triangle area question we generally need to know the base and height of the triangle. In this question you can extend the base AC to the point D and then draw a perpendicular from C to D to get the height CD. Refer below figure for detailed solution -

Attachment:
Triangle.jpg
Triangle.jpg [ 47.24 KiB | Viewed 621 times ]
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Re: what is the area of the triangle ABC?  [#permalink]

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New post 13 Mar 2018, 12:48
adkikani wrote:
pushpitkc niks18 Hatakekakashi
amanvermagmat Bunuel

Any way to link opposite sides with angles here since ratio between sides is given?

If AB = x, BC = 2x and AB + BC = 60. We can get AB = 20 and BC = 40.
How do I get third side length here?


brilliant explanation by niks18
Re: what is the area of the triangle ABC? &nbs [#permalink] 13 Mar 2018, 12:48
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