Prove these with detailed steps.

[Wong Siu-Yeung]

With detailed steps, prove ALL the following.
1. Given the side length of a square equals n, the diagonal of the square must equal n times the square root of 2. With DETAILED steps, please prove that the above statement is correct given n is a real number larger than 0 and n does not equal 0.
2. If the length of a rectangle equals g, and the width of a rectangle equals y, please use detailed steps to give the diagonal of such a rectangle, given g and y are both real numbers larger than 0 and neither g nor y equals 0.
3. If the non-right-angles of a right-angled triangle both equals 45º, the ratio of the side lengths are 1 : 1 : sqrt(2). Prove this.
4. If the non-right angles of a right-angled triangle equals 30º and 60º respectively, the ratio of the side lengths are 1 : sqrt(3) : 2. Prove this.
5. If the length, width and height of a cuboid are l,w and h respectively, the diagonal of such a cuboid is the square root of the sum of l², w² and h². Prove this.
6. If the side length of a cube is a, the diagonal of this cube equals a times sqrt(3). Prove this.

 

[Dr.Peterson]

With detailed steps, prove ALL the following.
1. Given the side length of a square equals n, the diagonal of the square must equal n times the square root of 2. With DETAILED steps, please prove that the above statement is correct given n is a real number larger than 0 and n does not equal 0.
2. If the length of a rectangle equals g, and the width of a rectangle equals y, please use detailed steps to give the diagonal of such a rectangle, given g and y are both real numbers larger than 0 and neither g nor y equals 0.
3. If the non-right-angles of a right-angled triangle both equals 45º, the ratio of the side lengths are 1 : 1 : sqrt(2). Prove this.
4. If the non-right angles of a right-angled triangle equals 30º and 60º respectively, the ratio of the side lengths are 1 : sqrt(3) : 2. Prove this.
5. If the length, width and height of a cuboid are l,w and h respectively, the diagonal of such a cuboid is the square root of the sum of l², w² and h². Prove this.
6. If the side length of a cube is a, the diagonal of this cube equals a times sqrt(3). Prove this.

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We don't give detailed steps; that's your task. Our role is to try to help you work it out; and we do that by starting where you are. To do that, we need to see what you can do, and where you need help.

None of these are very hard; what counts as "detailed steps" depends on what theorems you have learned, and how proofs are done in your course.

So, what can you show us?