Pls can you help me answer all of them
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Yes, this is a typical pythagorean triad.
Since the theorem is:
And by inspection,
would be the hypotenuse,
25 = 9 + 16
RHS = 9 + 16 = 25 = LHS
Therefore, we have proved that the values fit into pythagoras' theorem, which means that the triangle is a right triangle (that is the central idea of this theorem).
Since the theorem is:
And by inspection,
25 = 9 + 16
RHS = 9 + 16 = 25 = LHS
Therefore, we have proved that the values fit into pythagoras' theorem, which means that the triangle is a right triangle (that is the central idea of this theorem).
The distance 
between 
and 
is:
Since the equation of the hyperbola is
, we can get 
by itself and end up with
which we can plug into our distance formula:
To make calculation easier, we'll square both sides:
and create a new variable
:
Differentiate both sides:
Minimum distance is achieved when
:
To find a value of
, you can use methods like synthetic division and get the answer 
Plug into:
So the closest point on the hyperbola to is 
Since the equation of the hyperbola is
which we can plug into our distance formula:
To make calculation easier, we'll square both sides:
and create a new variable
Differentiate both sides:
Minimum distance is achieved when
To find a value of
Plug into
So the closest point on the hyperbola to