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655-705 Level|   Geometry|      
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Bunuel
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In a right triangle ABC, if the length of AB is 5 units and angle ACB 19 Jan 2022, 07:30
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In a right triangle ABC, if the length of AB is 5 units and angle ACB = 90 degrees, then what is the perimeter of the triangle?

(1) Area of triangle ABC = 6 sq. units
(2) The length of all the sides is an integer
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In a right triangle ABC, if the length of AB is 5 units and angle ACB 05 Feb 2022, 22:33
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Bunuel
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In a right triangle ABC, if the length of AB is 5 units and angle ACB = 90 degrees, then what is the perimeter of the triangle?

(1) Area of triangle ABC = 6 sq. units
(2) The length of all the sides is an integer


Bumping!!! Knockout this one and get KUDOS for a correct solution!!!

Let the two sides AC and CB be equal to a units and b units. We know that since it is a right triangle, a² + b² = 25

Quote:
(1) Area of triangle ABC = 6 sq. units
since area of triangle ABC here is ½*AC*BC = ½*a*b = 6 which means a*b=12

a*b=12 and a²+b²=25, substituting b=12/a into the second eq. we get

a² + 144/a² = 25
a⁴ - 25a² + 144 = 0
(a²-16)*(a²-9)=0
a² = 16 or a² = 9
a = 4 or a = 3
b = 3 or b = 4

In either case the sum a+b=7 is the same and when added with 5 (AB), gives us the perimeter of the triangle. Statement 1 is sufficient. Eliminate options B, C, and E.

Quote:
(2) The length of all the sides is an integer
this means that only a²+b²=25 but a and b are integers too and since we know that the hypotenuse is the largest side of a right angled triangle, both a and b less than 5, or can be any of {1, 2, 3, 4}

Using combinations we arrive at the result that only 3²+4² will fetch us 25. Hence, a=3/4 and b=4/3 and a+b+5 = 12 (perimeter of the triangle) Statement 2 is also sufficient.

Hence, the question can be answered by using either statement alone, option D.

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