Expert Guidance for Your Pythagorean Theorem Questions

Two congruent circles with centres at (2,3) and (5,6), which intersect at right angles, have radius equal to?

State whether the given statement is true or false: 9, 40, 41 is a Pythagorean triplet. True or false

5, 12, 13 is a Pythagorean triplet. True or false

8, 15, 17 is a Pythagorean triplet

Based on Pythagorean identities, which equation is true?
A. ${\sin }^2<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->1={\cos }^2<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}$
B. ${\sec }^2<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->{\tan }^2<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}=-<!--\; -\; -->1$
C. -${\cos }^2<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->1={\sin }^2<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}$
D. ${\cot }^2<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->{\csc }^2<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}=-<!--\; -\; -->1$

The end rollers of bar AB(1.5R) are constrained to the slot. If roller A has a downward velocity of 1.2 m/s and this speed is constant over a small motion interval, determine the tangential acceleration of roller B as it passes the topmost position. The value of R is 0.5 m.

The area of the obtuse angle triangle shown below is:
A17.5 sq. units
B 25 sq. units
C 14 sq. units
D 15 sq. units

Replacing x with $\frac{1}{2}$ in $2x^2$ will give you an answer of 1.

Why does the Pythagorean theorem only work for right triangles?

7, 24, 25 is a Pythagorean triplet.
A.True
B.False

A bicycle wheel with a 5 inch ray rotates ${60}^{\circ <!--\; \circ \; -->}$. What distance has the bicycle traveled?

The legs of a right triangle are 6 and 8 cm. Find the hypotenuse and the area of ​​the triangle.

Solve for X.
Anlge A=8x+5
Angle B=4$x^2$-10
Angle C=$x^2$+2x+10
I know that they equal 180 degrees. However I am drawinga blank on the factoring part of it

For each of the following, can the measures represent sides ofa right triangle? Explain your answers.a. 3 m, 4 m, 5 mb. $\sqrt{2cm},\sqrt{3cm},\sqrt{5cm}$

A 10-m ladder is leaning against a building. The bottom of theladder is 5-m from the building. How high is the top of theladder?

Pythagorean theorem and its cause
I'm in high school, and one of my problems with geometry is the Pythagorean theorem. I'm very curious, and everything I learn, I ask "but why?". I've reached a point where I understand what the Pythagorean theorem is, and I understand the equation, but I can't understand why it is that way. Like many things in math, I came to the conclusion that it is that way because it is; math is the laws of the universe, and it may reach a point where the "why" answers itself. So what I want to know is, is there an explication to why the addition of the squared lengths of the smaller sides is equal to the squared hypotenuse, or is it just a characteristic of the right triangle itself? And is math the answer to itself?
Thank you.

pythagorean theorem extensions
are there for a given integer N solutions to the equations
$\underset{n=1}{\overset{N}{\sum <!--\; \sum \; -->}}{x}_i^2=z^2$
for integers $x_i$ and zan easier equation given an integer number 'a' can be there solutions to the equation
$\underset{n=1}{\overset{N}{\sum <!--\; \sum \; -->}}{x}_i^2=a^2$
for N=2 this is pythagorean theorem

"Pythagorean theorem" for projection onto convex set
I'm going through the book on online convex optimization by Hazan, and in the first chapter I saw this assertion (which Hazan calls the "pythagorean theorem"):
Let $K\subset <!--\; \subset \; -->{\mathbb{R}}^d$ be a convex set, $y\in <!--\; \in \; -->{\mathbb{R}}^d$, and $x={\mathrm{\Pi <!--\; \Pi \; -->}}_K(y)$. Then for any $z\in <!--\; \in \; -->K$ we have:
$\Vert <!--\; \Vert \; -->y-<!--\; -\; -->z\Vert <!--\; \Vert \; -->\ge <!--\; \ge \; -->\Vert <!--\; \Vert \; -->x-<!--\; -\; -->z\Vert <!--\; \Vert \; -->.$
It is presented without proof - what is a proof for this? Also, how does it relate to the pythagorean theorem?

Non-geometric Proof of Pythagorean Theorem
Is there a purely algebraic proof for the Pythagorean theorem that doesn't rely on a geometric representation? Just algebra/calculus. I want to TRULY understand the WHY of how it is true. I know it works and I know the geometric proofs.

The Pythagorean theorem and Hilbert axioms
Can one state and prove the Pythagorean theorem using Hilbert's axioms of geometry, without any reference to arithmetic?
Edit: Here is a possible motivation for this question (and in particular for the "state" part of this question). It is known that the theory of Euclidean geometry is complete. Every true statement in this theory is provable.
On the other hand, it is known that the axioms of (Peano) arithmetic cannot be proven to be consistent. So, basically, I ask if there is a reasonable theory which is known to be consistent and complete, and in which the Pythagorean theorem can be stated and proved.
In summary, I guess I am asking - can we be sure that the Pythagorean theorem is true? :)