Question

a certain right angled triangle has its area numerically equal to its perimeter. the length of each side is an even integer. What is the perimeter

Answer 1

We use the concept of Pythagorean triplets.

Firstly: area=perimeter (numerically)

let perpendicular sides be a and b
so
$c = \sqrt{a^2 + b^2}$

Now;
$(1 \div 2)ab = a + b + c$
$= > (1 \div 2)ab = a + b + \sqrt{ {a}^{2} + {b}^{2} }$
Now,
${c}^{2}$
has to be a perfect square

Using values of Pythagorean triplets for a and b, we find that 5 and 12 satisfy the given equation..

Therefore,minimum perimeter = a+b+c= 5+12+13= 30

Answer 2

Given that, the measure of each side of the right triangle is an even integer unit.

Also, its perimeter and area are numerically equal.

Let us start from even integers: 2,4,6,8,10,12,14, etc.

Out of them, only (6,8,10) make Pythagorean triplet

∵8  

2

+6  

2

=10  

2

 

Also, 8+6+10=24(Perimeter)

And,  

2

1

​  

×8×6=24(Area)

∴ This group of even integers comply with all the given conditions.

∴ Perimeter =24 units.

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