The answer is: " 1.4142 ft " .
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Explanation:
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Note: All four (4) sides of a square have the same length.
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The "diagonal" of a square forms a right triangle, in which the "diagonal" is the hypotenuse; and the two (2) sides of the square are the other two (2) angles.
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So, we have:
| \
1 ft | \ c
| \
|_ \
| _| _ _ _ _\
1 ft.
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Solve for "c" ; the "hypotenuse" .
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Use the Pythogorean theorem (since this is a "right triangle") ; to solve for "c" ;
a² + b² = c² ;
Rearrange the equation, as follows:
↔ c² = a² + b² ;
in which: a = the length of one side of the triangle = 1 (given) ;
b = the length of the other side of the triangle = 1 (given) ;
c = the length of the hypotenuse of the triangle ;
→ for which we wish to solve;
Given: c² = a² + b² ;
Plug in our known values for "a" and "b" ; to solve for "c" ;
→ c² = 1² + 1² ;
= (1 * 1) + (1 * 1) ;
= 1 + 1 ;
→ c² = 2 ;
Now, take the "positive square root" of each side of the equation ;
to isolate "c" on one side of the equation ; & to solve for "c" ;
→ +√(c²) = +√2 ;
to get: → c = √2 ft.; or write as: " 1.414213562373095 ft." ;
→ round to: "1.4142 ft."
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Other:
Note: The "diagonal" of a square forms two right (2) triangles. Each triangle is a " 45°-45°-90° " triangle.
Note that the "diagonal" of such a triangle is the hypotenuse; and the length of each of two sides of such a triangle will be the same; (since all 4 (four) side lengths of a square are the same. For such: "45°-45°-90° " triangles, The lengths of the sides take the form: "a, a, and "a√2" ; in which "a" and a" (which are equal, of course) represent the 2 sides of the triangle, and "a√2" represents the hypotenuse.
This is consistent with our calculations:
a = 1 ; a = 1 ; and "a√2" — which is the hypotenuse—is: "1√2" = "√2" .
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