Is the Pythagorean Theorem Applied to Determine the Length of AB?

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SUMMARY

The length of segment AB is definitively determined to be 6 using the Pythagorean Theorem. The coordinates of point A are (0, 4 + √2) and point B are (4 - √2, 0). The calculation involves applying the distance formula, which simplifies to √((4 - √2 - 0)² + (0 - (4 + √2))²). The final result confirms that the hypotenuse of the triangle formed by these points is indeed 6, as derived from the expression 2(16 + 2) = 36.

PREREQUISITES
  • Understanding of the Pythagorean Theorem
  • Familiarity with the distance formula in coordinate geometry
  • Basic knowledge of square roots and algebraic manipulation
  • Ability to interpret and work with coordinates in a Cartesian plane
NEXT STEPS
  • Study the derivation and applications of the Pythagorean Theorem in various geometric contexts
  • Learn advanced techniques for simplifying expressions involving square roots
  • Explore the distance formula in three-dimensional space
  • Investigate the properties of triangles and their applications in real-world scenarios
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Students studying geometry, mathematics educators, and anyone interested in applying the Pythagorean Theorem to solve geometric problems.

Ismael Nunez
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Anyone want to take a crack at it? My class has been discussing it: Find the length of AB:http://t4.rbxcdn.com/84e25f3830d66e6bbaeaba48e35c0781
 
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Six.
 
Misha Kuznetsov said:
Six.
Will you please elaborate?
 
6?
 
Jerry Friedman said:
6?
Ok, great, but how did you get there?
 
I am probably wrong, but I got sqrt(34).
 
Find the coordinates of point A and use the distance formula to get AB. There are pleasant cancellations.
 
I just found the hypotenuse of a triangle with legs 4+sqrt(2) and 4-sqrt(2).
 
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Jerry Friedman said:
Find the coordinates of point A and use the distance formula to get AB. There are pleasant cancellations.
I did, and I still got sqrt(34)... One question though, the square at the top left corner... If split vertically, to get 4 triangles, wouldn't the legs of one of those triangles be one?
 
  • #10
No, each leg would be sqrt(2) .
 
  • #11
Misha Kuznetsov said:
No, each leg would be sqrt(2) .
Alright, I see my mistake, I checked my work again. Thanks.
 
  • #12
Yep, no problem. :smile:
 
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  • #13
First, we choose the axes so as to simplify our calculation. Let us drop a perpendicular from A to the base and consider the the y-axis.

The location of A is (0, 4 +√²) and B is (4 - √2, 0)
Distance is √x²+y²

As we know, (a - b)² + (a + b)² = 2( a² + b²)

Which here is 2( 16 +2) = 36, the square root of it is 6.

I'm sorry if I skipped some steps. It's very difficult to type in mathematical notation.
 
  • #14
Was it given that the figures with sides 2 and 4 are squares?

If so, 6.
 
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