The leg of a right triangle is equal to 1/5 the sum of the other sides

Question

The leg of a right triangle is equal to 1/5 the sum of the other sides. If the perimeter of of the triangle equals 1, what is its area?

A. 1/60
B. 1/50
C. 1/45
D. 1/30
E. 1/15

Kinshook · Answer

Given: The leg of a right triangle is equal to 1/5 the sum of the other sides. 
Asked: If the perimeter of of the triangle equals 1, what is its area?

Let the legs od right triangle be x & y and hypotenuse be z

x^2 + y^2 = z^2   (1) 
 x = \frac{1}{5 }(y + z)   (2)
x + y + z = 1 (3)

(2) + (3)
 \frac{1}{5}(y+z) + y + z = 1 
 y + z = \frac{5}{6}   (4)

x = \frac{1}{6}   (5)

\frac{1}{36 } = z^2 - y^2 = (y+z)(z-y) = \frac{5}{6} (z-y) 
 z - y = \frac{1}{30}   (6)

(4) + (6)
 2y = \frac{5}{6 }- \frac{1}{30} = \frac{24}{30} = \frac{4}{5} 
 y = \frac{2}{5} = 0.4  (7)

Area = \frac{1}{2} * x * y = \frac{1}{2} * \frac{1}{6} * \frac{2}{5} = \frac{1}{30}

IMO D

Babineaux · Answer

Please refer to the pic uploaded.

IMO ans is D.

Anyone please explain how the problem can be solved in a simpler way because this process is time consuming one.

Posted from my mobile device

GMATinsight · Answer

Leg lenth = x
Sum of the other two sides = 5x

Let, second Leg = a

i.e.  x^2+a^2 = (5x-a)^2

Also,  6x = 1 
i.e.  x = \frac{1}{6}

Now, i.e.  (\frac{1}{6})^2+a^2 = (\frac{5}{6}-a)^2

i.e.  \frac{1}{36} + a^2 = \frac{25}{36} +a^2 - \frac{5a}{3}

i.e.  \frac{5a}{3} = \frac{24}{36} = \frac{2}{3}

i.e.  a = \frac{2}{5}

Area = (\frac{1}{2})*x*a = (\frac{1}{2})*(\frac{1}{6})*(\frac{2}{5}) = \frac{1}{30}

Answer: Option D

ScottTargetTestPrep · Answer

Solution:

Even though the perimeter is 1, the right triangle in question is probably just a special right triangle that is related to the 3-4-5 right triangle or the 5-12-13 right triangle, etc.

Let's say it’s a 3-4-5 right triangle. We see that the shortest side is 3, which is 1/3 of the sum of the other two sides (4 + 5 = 9). So it’s not a 3-4-5 right triangle.

Now, let’s say it’s a 5-12-13 right triangle. We see that the shortest side is 5, which is indeed 1/5 of the sum of the other two sides (12 + 13 = 15). So it IS a 5-12-13 right triangle. Since the perimeter of the triangle is 1, we can create the equation:

5x + 12x + 13x = 1

30x = 1

x = 1/30

So the three sides of the triangle are 5/30 = 1/6, 12/30 = ⅖, and 13/30. Since the area of a triangle is ½ the product of the two legs, the area of the triangle is:

1/2 x 1/6 x 2/5 = 1/30

Alternate Solution:

Let x be the length of the leg that equals 1/5 of the sum of the remaining sides. Then, the sum of the remaining sides is 5x,and since the perimeter is 1, we have:

x + 5x = 1

6x = 1

x = 1/6

Let y be the other leg and z be the hypotenuse. Then, we have:

1/6 + y + z = 1

z = 1 - 1/6 - y

z = 5/6 - y

Let’s square each side of this equality:

z^2 = 25/36 - 5y/3 + y^2

Notice that we have (1/6)^2 + y^2 = z^2 using the Pythagorean theorem, which simplifies to z^2 = y^2 + 1/36. Let’s substitute this into the equality above:

y^2 + 1/36 = 25/36 - 5y/3 + y^2

5y/3 = 24/36 = 2/3

y = 2/5

Thus, the area of the triangle is (1/6)*(2/5)*(1/2) = 1/30.

Answer: D

The leg of a right triangle is equal to 1/5 the sum of the other sides

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The leg of a right triangle is equal to 1/5 the sum of the other sides

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