... COORDINATE GEOMETRY Find the measures of the sides of ABC and classify
each triangle by its sides.
17. A(5, 4), B(3, -1), C(7, -1)
18. A(-4, 1), B(5, 6), C(-3, -7)
19. A(-7, 9), B(-7, -1), C(4, -1)
... 8. Wheel A has four lines of reflectional symmetry;
Wheel C has five lines of reflectional symmetry.
Wheels B and D do not have reflectional symmetry.
9. Wheels B and D have only rotational symmetry.
Wheels A and B have 4-fold, Wheel C has 5-fold,
and Wheel D has 3-fold rotational symmetry.
10, 11. ...
... Based on the following details, are the triangles definitively congruent?
6. Both triangles are right triangles in which one angle measures 55◦ . All of their corresponding sides are
7. Both triangles are equiangular triangles.
8. Both triangles are equilateral triangles. All sides are 5 ...
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Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g. systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law.