Find Side Lengths of Pythagorean Triangle ABC with Square DEFG Inscribed

In summary, to find the length of side AB, use the Pythagorean theorem by setting up the equation AB^2 = DE^2 + EF^2 and solving for AB. Any side of the inscribed square can be used to find the triangle's side lengths, but using the longer sides will result in a more accurate measurement. A triangle is a Pythagorean triangle if the square of the longest side is equal to the sum of the squares of the other two sides. The Pythagorean theorem can only be used for right triangles, and a side length of 0 for the inscribed square is not possible as it would result in a degenerate triangle.
  • #1
Wilmer
307
0
Pythagorean triangle ABC with area 8214
has square DEFG with sides = 60 inscribed in it.
Side DE of the square lies on the hypotenuse.
Find the triangle's side lengths.
Code: 
B

         E                     D

F
C         G                               A
 
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  • #2
In case I lose it!
a = BC, b = AC, c = AB, d = square side
Formula:
d = abc / (a^2 + b^2 + ab)

Solution: a = 111, b = 148, c = 185

111 * 148 / 2 = 8214
 
  • #3
Wilmer said:
Formula:
d = abc / (a^2 + b^2 + ab)

Where did you get it?
 
  • #4
Monoxdifly said:
Where did you get it?
Made it up...
 
  • #5
Duh...
 
  • #6
Monoxdifly said:
Duh...
Deaf Upon Hearing ?
 
  • #7
Wilmer said:
Deaf Upon Hearing ?

Dumb Upon Hearing
 

1. How do you determine the side lengths of a Pythagorean triangle with a square inscribed inside?

To determine the side lengths of a Pythagorean triangle with a square inscribed inside, you can use the Pythagorean theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the sides of the square (DEFG) will be the legs of the right triangle, and the diagonal of the square will be the hypotenuse.

2. What is a Pythagorean triangle?

A Pythagorean triangle is a right triangle where the lengths of all three sides are positive integers. It follows the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

3. How do you find the length of the diagonal of a square inscribed inside a Pythagorean triangle?

To find the length of the diagonal of a square inscribed inside a Pythagorean triangle, you can use the Pythagorean theorem. The diagonal of the square will be the hypotenuse of the right triangle formed by the sides of the square. Use the lengths of the two legs (which are also the side lengths of the square) to find the length of the hypotenuse.

4. Can a Pythagorean triangle have non-integer side lengths?

No, a Pythagorean triangle must have all integer side lengths. This is because the Pythagorean theorem only holds true for right triangles with integer side lengths. If the side lengths were non-integer, the equation would not be balanced and the theorem would not hold.

5. How do you know if a triangle is a Pythagorean triangle?

You can determine if a triangle is a Pythagorean triangle by using the Pythagorean theorem. If the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides, then the triangle is a Pythagorean triangle. Additionally, all three side lengths must be positive integers for a triangle to be considered a Pythagorean triangle.

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