Integer-Sided Right Triangle: 2001 Leg Length & Minimum Other Leg Length

In summary, an integer-sided right triangle is a triangle with three sides that are all whole numbers and one of the angles is a right angle. The leg lengths of the 2001 integer-sided right triangle are 2001 and 2672, as calculated by the Pythagorean theorem. The minimum length for the other leg of an integer-sided right triangle is always 1, and an integer-sided right triangle can have equal leg lengths, known as an isosceles right triangle. These triangles have practical applications in fields such as architecture, engineering, and surveying for calculating distances, angles, and heights.
  • #1
lfdahl
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The shorter leg of an integer-sided right triangle has length 2001. How short
can the other leg be?
 
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  • #2
lfdahl said:
The shorter leg of an integer-sided right triangle has length 2001. How short can the other leg be?
my solution:
$2001=667\times 3=3\times 23\times 29$
so the other leg =$667\times 4=2668$
 
Last edited:
  • #3
Thankyou Albert for the correct answer - and for your tireless participation!(Clapping)

Suggested solution:

Let $a,b,c$ be the sides of the triangle. Thus $2001 = a < b < c$. Set $c=b+m$. Then $(b+m)^2 = b^2+2001^2$ or $m(2b+m) = 2001^2$. So $m$ is a divisor of $2001^2=3^2\cdot 667^2$ and since $b= c-m$ is to be shortest ($> 2001$), $m = 667$ (the next largest divisor is $3 \cdot 667 = 2001$, which makes $b=0$) should be considered. Then $667(2b+667) = 9\cdot667^2$ gives $b=2668$ and $c=2668 + 667=3335$. One checks, that $2001^2+2668^2 = 3335^2$.
Comment: This triangle is the $(3,4,5)$ triangle since $(2001,2668,3335) = 667(3,4,5)$. But recognizing this, does not prove, that $2668$ is the shortest possible side larger than $2001$.
 

FAQ: Integer-Sided Right Triangle: 2001 Leg Length & Minimum Other Leg Length

What is an integer-sided right triangle?

An integer-sided right triangle is a triangle with three sides that are all whole numbers, and one of the angles is a right angle (90 degrees).

What are the leg lengths of the 2001 integer-sided right triangle?

The leg lengths of the 2001 integer-sided right triangle are 2001 and 2672, as calculated by the Pythagorean theorem (a^2 + b^2 = c^2).

Is there a minimum length for the other leg of an integer-sided right triangle?

Yes, the minimum length for the other leg of an integer-sided right triangle is always 1. This is because the Pythagorean theorem states that the square of the hypotenuse (the longest side) must be equal to the sum of the squares of the other two sides. Therefore, the other leg cannot be less than 1, as it would result in a negative length.

Can an integer-sided right triangle have equal leg lengths?

Yes, an integer-sided right triangle can have equal leg lengths, as long as they are whole numbers and one of the angles is a right angle. This is known as an isosceles right triangle.

Are there any practical applications for integer-sided right triangles?

Yes, integer-sided right triangles have many practical applications in fields such as architecture, engineering, and surveying. They can be used to calculate distances, angles, and heights in real-world scenarios.

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