What is the area of triangle ABC above, with side lengths x,y, and z?

Area of triangle = 1/2*x*y

Statement 1=(x+y)^2−(x−y)^2= (x^2+y^2+2xy)- (x^2+y^2-2xy)=4xy=80

4xy=80
1/2*x*y=10
Sufficient

Statment 2 x-y=1
We can't find x*y with the given information
Insufficient

A

What is the area of triangle ABC above, with side lengths x, y, and z?

(1) (x+y)^2−(x−y)^2=80
(2) (x−y)=1

Area of right angled triangle = xy/2

Statement 1 gives
(x+y)^2-(x-y)^2 = 80
(x^2+y^2+2xy) - (x^2+y^2-2xy) = 80
4xy = 80
xy/2 = 10 = Area of the triangle
Statement 1 Sufficient.

Statement 2 gives
x-y = 1
It will provide infinite possibilities of x, y & z
Statement 2 Insufficient.

IMO A

Solution:

Question Stem analysis:

Triangle ABC is a right angled triangle making AB represented as x as it's height, and BC represented by y as it's base. Area of the triangle is 1/ X Base X Height Therefore we need the values of x & y to identify he area.

Statement one alone:

(x + y)^2 - (x - y)^2 = 80, when we solve this, we get 4xy = 80 & xy=20 Hence we can find out the area of the triangle which comes to 10
We can safely eliminate choices B,C & E.

Statement 2 alone:

(x - y)= 1
Clearly, this statement is not sufficient to find out the values of x & y, Thus statement two alone is not sufficient.
Hence the answer is A

Question: What is the area of triangle?

Observations:
The given image is a right angled triangle with the sides x,y,z. Consider, base = y, and height = x
Area of Triangle = (1/2)*base*height = (x*y)/2

Statement 1: (x+y)\(^2\)−(x−y)\(^2\)=80. Simplifying this,
x\(^2\)+y\(^2\)+2xy-x\(^2\)-y\(^2\)+2xy = 80
4xy = 80. From this we can easily find (x*y)/2 which is the area of the triangle.
Remember we don't need to find the exact value in DS problems, we just need to have one true answer and we have it.
Therefore Statement 1 is sufficient. ---> AD/BCE

Statement 2: (x−y)=1
In this case, x and y can take more than one single value and still make the statement true. In this case, we don't need to find the value of z even though it is easy because we don't need it.

For example,
when x=4, y=3 -> Area of triangle = 6
when x=5, y=4 -> Area of triangle = 10.

Therefore, Statement 2 is insufficient

AD/BCE

The correct answer choice is A

DS Question (Geometry):

Area of a triangle = 1/2 * base * height

For given figure, area = 1/2 * y * x

Option 1:
(x + y)^2 - (x - y)^2 = 80
x^2 + y^2 + 2xy - x^2 - y^2 + 2xy = 80
4xy = 80
xy = 20

Therefore using the value of xy we can find area of given figure.
Hence option is sufficient.

Option 2:
(x - y) = 1
x = y + 1

There can be multiple right angled triangles formed where the base and height have a difference of just 1

From our knowledge of pythagorean triplets we know
3,4,5
20,21,29
are valid pythagorean triplets.

Hence Option 2 is not sufficient.

Answer: A

We want to find the value of the area of right triangle ABC = \(\frac{1}{2}*x*y\)
So, knowing the values of both x and y (or) knowing the product of x and y will be sufficient.

(1) \((x+y)^2 – (x-y)^2 = 80\)
=> \(x^2 + 2xy + y^2 – [x^2 – 2xy + y^2] = 80\)
=> \(x^2 + 2xy + y^2 – x^2 + 2xy – y^2 = 80\)
=> \(4xy = 80\)
=> \(xy = 20\)

At this point, we can say (1) is sufficient.

(2) (x-y) = 1
=> \(x = y + 1\)

Substituting the above value of x in the Pythagoras equation: \(x^2 + y^2 = z^2\) we get
\((y+1)^2 + y^2 = z^2\)
=> \(y^2 + 2y + 1 + y^2 = z^2\)
=> \(2y^2 + 2y + 1 = z^2\)

Now we have the following values:
\(y = y\)
\(x = y + 1\)
\(z^2 = 2y^2 + 2y + 1\)

Let, y=2
Therefore, \(x = y+1 = 3\) ; \(x^2 = 9\) ; \(y^2 = 4\)
\(z^2\) = 2(4) + 2(2) + 1 = 13
Therefore, (2 , 3 , \(\sqrt{13}\)) is one possible triangle.

Let, y = 3
Therefore, \(x = 4\) ; \(z^2 = 25\)
Thus, (3, 4, 5) is another possible triangle which satisfy the condition.
Hence, (2) is Insufficient.

Answer (A)

What is the area of triangle ABC above, with side lengths x, y, and z?


(1) \((x+y)^2−(x−y)^2=80\)

(2) \((x−y)=1\)

Question:(!/2)*x*y = ? i.e. x*y = ?

Statement 1: \((x+y)^2−(x−y)^2=80\)
i.e. x^2 + y^2 + 2xy - x^2 - y^2 + 2xy = 4xy = 80
i.e. x*y = 40 hence area = 40/2 = 20
SUFFICIENT

Statement 2: \((x−y)=1\)
x and y may be 4 and 3 or 5 and 4 respectively hence inconsistent value of area
NOT SUFFICIENT

Answer: Option A

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