A piece of paper in the shape of an isosceles right triangle is cut...

Question

A piece of paper in the shape of an isosceles right triangle is cut along a line parallel to the hypotenuse of the triangle, leaving a smaller triangular piece. If the area of the triangle was 25 square inches before the cut, what is the new area of the triangle?

(1) The cut is made 2 inches from the hypotenuse

(2) There was a 40 percent decrease in the length of the hypotenuse of the triangle

(1) The cut is made 2 inches from the hypotenuse

(2) There was a 40 percent decrease in the length of the hypotenuse of the triangle

chetan2u · Answer

What we have are two similar triangles -- ADE, the initial one, and ABC, after the cut.

The area of two similar triangles is in the ratio of the squares of their sides...
The area of ADE =  25....\frac{1}{2}*AE^2=25....AE=5\sqrt{2}

(1) The cut is made 2 inches from the hypotenuse
Take triangle COE, which is again isosceles right angled triangle..CO=OE=2, so CE= \sqrt{2^2+2^2}=\sqrt{8}=2\sqrt{2} 
Thus the side AC= 5\sqrt{2}-2\sqrt{2}=3\sqrt{2} 
Ratio of sides =  5\sqrt{2}:3\sqrt{2} , 
so Area will be in the ratio  (5\sqrt{2})^2:(3\sqrt{2})^2=50:18=25:9 
New area =  25*\frac{9}{25}=9

(2) There was a 40 percent decrease in the length of the hypotenuse of the triangle
So sides are in the ratio =  x:0.6x=10:6=5:3 ...
so Area will be in the ratio  (5\sqrt{2})^2:(3\sqrt{2})^2=50:18=25:9 
New area =  25*\frac{9}{25}=9

D

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • Triangle is an isosceles right-angle triangle
 o Base = perpendicular = x(say)
o Area =  \frac{1}{2} *x*x 
o  25 = \frac{1}{2}*x^2 
   x = \sqrt{50} =5\sqrt{2}  
o Hypotenuse = 10  
We need to find the area of the triangle after the cut.
 o We need to find any of the side after the cut.
  After all it's goining to be an isosceles right-angled triangle only  
 Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  The cut is made 2 inches from the hypotenuse.
 https://www.dropbox.com/s/92gok47flflgjy7/43.jpg?dl=1 
 • Now, simply BD is a perpendicular of AC, and it will divide the AC into two equal parts
 o BDC is a right-angled triangle
   BC = 5*\sqrt{2}, CD = 5   
•  BD = 5 
 o Now,  BM = 5-2 = 3 
   EM =MF = BM = 3   
 Now,  EF = EM + MF = 3+ 3 = 6 
 This is the hypotenuse of new triangle. 
o Thus, we can find the area.  
Hence, statement 1 is sufficient, we can eliminate the answer options B, C, and E.
 Statement 2:  There was 40 percentages decrease in the length of the hypotenuse.
 • New hypotenuse =  10 – \frac{10}{100}*40 = 10 – 4 = 6 
 o Thus, we can find the area of new triangle.  
Hence, statement 2 is also sufficient, the correct answer is  Option D.

Victor19 · Answer

The answer is D.

We know it is an isosceles triangle and area = 25 sq cm. 
This means 1/2 * base * height = 25 sq cm ---- base^2 = 50 sq cm ----- base = 5*sqrt(2).

Statement 1 : The cut is made 2 inches from the hypotenuse.
Since this is parallel to the hypotenuse, the base and height are going to decrease in length by the same amount. No need to calculate. We can find the info using the property of similar triangles.
Sufficient.

Statement 2 : There was a 40 percent decrease in the length of the hypotenuse of the triangle.
Again, this is similar to the first statement and using the same strategy we can find the values.
Sufficient.

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