In the trapezoid ABCD, what is the area of triangle ABC?

Question

https://gmatclub.com/forum/download/file.php?id=29101 
In the trapezoid ABCD, what is the area of triangle ABC?

(1) AC = 10
(2) BC = 7

Skywalker18 · Answer

We need to find Area of  	riangle  ABC 
(1) AC = 10
In  	riangle  ADC ,
AD=8 
CD=6
and AC = 10
=>  \angle  ADC = 90
 	riangle ADC is right angled.
But we have no information about length of base BC
Not sufficient

(2) BC = 7
we have no information about height.

Not sufficient 
Combining 1 and 2 , we get
Since ABCD is a trapezoid and AD ≠BC
=> BC// AD , CD is height for triangle ABC as well.
Area of   	riangle ABC = 1/2 * base * height
= 1/2 * 7 * 6 = 21
Sufficient

Answer C

KS15 · Answer

Can you tell me how CD is the height for triangle ABC as well?

Skywalker18 · Answer

Hi KS15 , Because BC and AD are the two parallel sides of the trapezoid and the distance between the two parallel lines remains the same. In riangle ABC , we need to extend base BC and drop a perpendicular from vertex A . Now in this figure AE=CD = altitude of riangle ABC . Hope it helps !!

Skywalker18 · Answer

Alternatively , this question can also be solved as 
Area of  	riangle  ABC = Area of Trapezoid ABCD - Area of  	riangle  ADC
= 1/2 * Sum of bases * height - Area of  	riangle  ADC
= 1/2 *( BC +AD ) * CD - 1/2 * AD * CD

Answer C

KS15 · Answer

How did you assume the parallel sides? Its not given-correct?

RatneshS · Answer

I dont think that you can assume CD as height until it is given so. CD may not be a perpendicular
Area of trapezium =1/2(sum of parallel sides)*H
parallel sides are given but height is the missing information and we cannot find height from both the equations.
Answer should be E

Skywalker18 · Answer

Hi  KS15  ,

Since  \angle ADC = 90 
A trapezoid has a even number of right angles (Since atleast 2 sides of a trapezoid are parallel )
So, there are 2 cases possible - 
1.  \angle  BAD = 90
=> AB // CD ( If 2 lines are perpendicular to the same line then lines are parallel to each other )
But we are given that BC= 7 and AD = 8 => AD ≠BC
Distance between 2 parallel lines do not change .
Therefore , AB and CD are not parallel .
Our assumption that AB// CD is incorrect .
2.  \angle  DCB = 90
=> AD //BC

To simplify either AB // CD or AD//BC
But , since AD ≠BC
and since distance between 2 parallel lines do not change .
=>AD //BC

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

In the trapezoid ABCD, what is the area of triangle ABC?

(1) AC = 10
(2) BC = 7

In the original condition, the area of triangle ABC=area of trapezoid ABCD-the area of triangle ACD. Then, there are 3 variables(length of BC, BA, and AC), which should match with the number of equation. So you need 3 equations. For 1) 1 equation, for 2) 1 equation, which is likely to make E the answer. 
When 1)&2), since AC=10 and the angle D=90, the height should be 6 and BC=7, which is sufficient. Therefore, the answer is C.

-> For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

gauravategmat · Answer

@bunnel Please let me know how CD is considered height of trapezium ABCD. It definitely can be a slant line.

In the trapezoid ABCD, what is the area of triangle ABC?

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In the trapezoid ABCD, what is the area of triangle ABC?

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