Show that hypotenuse of right triangle is longer than any of its two legs?

Question

without invoking the 5th postulate.

That is do not assume that there is only one parallel line.

People, what Pythagoren thorem?

Answer 1

Well, I don't think that's possible because in spherical geometry, we can have a spherical right triangle where the side opposite to the right angle isn't the longest side. Unless there's some equivalent to the 5th postulate, we could be talking about spherical triangles.

All the other answerers above are using circular reasoning.

However, I will think about the case of Lobachevskian geometry, which is what you seem to be implying. After all, the limiting case in Euclidian geometry is where one of the sides approaches the length of the hypotenuse.

Allright, I'm going to borrow an "early" theorem from a text on Lobachevskian geometry to hurry up to the geometrical proof of this. The Saccheri-Legendre Theorem states that the sum of the interior angles of a triangle is always less than 180°. Next, any triangle with two equal sides will have equal base angles for symmetry reasons. Let us assume that we have a right Lobachevskian triangle which hypotenuse is LESS than the longer base. We pick a point along this base so that we can form an "isoceles" triangle, in which the base angles are supposed to be the same. Let A, B, and 90° be the angles of the original triangle, and x, y, and 90° be the angles of the triangle that is not the isoceles one. Then we have the following conditions which must be true:

A + B + 90 ≤ 180
x + y + 90 ≤ 180
A + (B - y) + (180 - x) ≤ 180
B - y = 180 - x (base angles are equal)
A + 360 - 2x ≤ 180
A + 180 ≤ 2x
But since x ≤ 90 - y, we know that x ≤ 90, therefore,
A ≤ 0, which is a contradiction.

Answer 2

The hypotenuse of an isosceles right triangle can not be 3 inches longer than the length of a leg... The length of the hypotenuse of an isosceles right triangle is always SQRT2 multiplied by the length of a leg (and the legs must be congruent). Therefore, it is impossible for the difference between the hypotenuse and leg to be a rational number (such as 3) since either the hypotenuse length is irrational or the legs are irrational in every case. PLEASE believe when I say you can't have a hypotenuse that is 3 inches longer than a leg. The difference in the hypotenuse and either leg MUST be irrational for every isosceles right triangle. I've included some references so you don't make the mistake of doing the algebra (as suggested by the others) a^2 + a^2 is never equal to (a + 3)^2

Answer 3

the pythagorean theorem is a^2+b^2=c^2. It states the sum of the squares of the shortest legs of a right triangle is equal to the square of the longest leg or hypotnuse.

If that doesn't answer your question, this should. The postulat stating that the shortest distance between a point and a line is a perpendicular line through the point is key. If side a is the line in this situation, and the point of the triangle that doesn't lie on line a (where the hypotnuse meets the other side) is the point, the shortest distance is a perpendicular line (perpendicular lines form RIGHT ANGLES) meaning the shortest distance is the other leg (side b) because it is perpendicular to the first side. The hypotnuse is not perpendicular to side a so it has to be longer than the shortest possible distance or (in this case) side b. You can prove it's longer than side a by saying side b is the line and where a meets the hypotnuse is the point. Like the other side, because side a is perpendicular to side b, it is the shortest distance between the two. The hypotnuse, again, is not perpendicular to the line so it is, therefore, longer than the shortest distance (side a).

Hope that makes sense!

Answer 4

In a triangle the longest side will be opposite the largest angle. Since the right angle is 90 degrees, the sum of the other two angles will be 90 degrees (all angles sum to 180 degrees in a triangle). Since an angle in the triangle has to be greater than zero the angles will have to be each less than 90 degrees. Therefore the hypotenuse will be the longest side.

Answer 5

The hypotenuse is the longest side in a triangle as there is another proof that the perpendicular is the shortest line that can be drawn to another line. Take a right triangle ABC and take b as the right angle. so here AB and BC are perpendiculars to each other. The only other side is the hypotenuse. hence it is the longest side. HENCE PROVED..... This is the perfect answer...

Answer 6

The Pythagoreon Theorem is a theorem that says the sum of the two legs (a and b) to the second power (²) will equal the length of the hypotenuse squared so that a²+b²=c². So if you need to add the squared lengths of the legs to get the length of the hypotenuse squared, the hypotenuse must be longer than either of the legs.

Answer 7

Just go back to Pythagoras's theorem. If you can find the hypotenuse of a right triangle by adding the square of each leg to equal the square of the hypotenuse, then by simple reasoning the hypotenuse will be longer then either leg because it is the indirect sum of both legs.

Answer 8

The hypotenuse of a right triangle will always be the diagonal of a rectangle formed by putting two of those triangles together, and that's always longer than the side of the rectangles.

Answer 9

According to the Pythagorean Theorem, a^2 = c^2 - b^2. If a^2 can be equal to c^2 - b^2 and b^2 = c^2 - a^2 it means that c^2 is greater than both of them since side length can only be positive.

Answer 10

Look up a proof of the Triangle Inequality, it will not only prove what you are looking for but is expanded into other vector spaces as well

http://en.wikipedia.org/wiki/Triangle_inequality
http://mathworld.wolfram.com/TriangleInequality.html