Why is the diagonal of a square longer than the sides?

ah yes it all makes sense now

Sometimes I can't tell if people are just trying to show off or if they genuinely think this would make sense to someone who doesn't even fucking know the Pythagorean theorem.
>hurr what i just posted was basically the Pythagorean theorem
Fuck off

>show off on anonymous Chinese cartoon forum.

Kek

It follows trivially from the trivial inequality

because you can't have an isosceles triangle with 2 right angles

Are you calling Pythagoras a liar, you little bitch?

In other words, if you infinitely extend the diagonal line, and one of the squares side lengths, and draw the remaining side length on as a line segment intersecting the two infinitely long lines, then you can apply the parallel postulate. That is, because the lines intersected on one side of the line segment, the sum of the angles on that side of the line segment must be less than 2 right angles. But, your triangle has to be isosceles, as you are assuming the diagonal and the side length are the same length, so the sum of the angles must be either angle, doubled. But, you know one of the angles is a right angle, so they are both right angles, so the sum is two right angles, which is impossible. By the parallel postulate, the lines could not have intersected, as the sum of the angles is exactly 2 right angles, so, the diagonal of the square cannot be equal to its sides, unless you change a bunch of euclidean geometry (such as the parallel postulate).

imho, which is kinda refected in other replies, because of how axioms work and shit STEM usually describes how things are more than it describes why they are. sorry for bad englando im kinda drunk

Triangle inequality.
A bit stubborn to prove, though.

Also there are different notions of "distance" you could use which would disagree.

en.wikipedia.org/wiki/Lp_space#The_p-norm_in_finite_dimensions

The shortest distance between two points is a straight line (the diagonal). Anything that diverges from that direct path (the sides) has to be longer

in philosophy they call this a'priori inductive reasoning, something that occurs because of its own definition, where the contradiction would be impossible.

its hard to answer the question "why" if it doesn't relate to any practical distinctions.
in euclidean geometry, the parallel postulate prevents the existence of a square with a diagonal the same length as one of its sides. In Spherical geometry, you can't make a square with a diagonal that is the same length as one of its sides (suppose you construct a square on the sphere with a diagonal the same length as one of the square's sides. cut off a part of the square with the diagonal as its boundary. the resulting shape is a triangle on a sphere. by assumption, the triangle's sides are all the same length. by assumption, each of the sides are "straight," in the context of the spherical surface. In other words, all of the sides follow the geodesics of the sphere. The only geodesics of the sphere are the great circles, so each side of the triangle rests on a great circle. They are all the same length, and all great circles on the sphere are the same, so the angles between the radii connecting the center of the sphere to the endpoints of the triangle's sides are all the same. By the rotational symmetry of these radii, we come to the conclusion that the angles of the triangle on the sphere must all be congruent. this is impossible, assuming the triangle was created by cutting a square in half: the triangle's angles can't all be equal. therefore, you could not have made a square on the sphere such that its diagonal is the same length as one of its sides). Its easier to come up with examples of situations where you can't make your square, than it is where you can, so its not a great question.

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[eqn]x^2 + y^2 = z^2[/eqn]