No.
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True, because the slant height and the altitude, or height, of the pyramid form one leg and the hypotenuse of a triangle withing the pyramid, and the hypotenuse of a triangle is always the longest side- it is not possible for the hypotenuse to be equal to the legs of a right triangle. (It is a right triangle because an altitude is perpendicular to the base of a pyramid.)
It is a modified version of the Pythagorean theorem instead of a^2+b^2=c^2 it is h^2+r^2= l^2 where h is height r is radius and l is slant height This is the same thing for slant height for pyramid except instead of radius, it is 1/2 the base
Yes there is a such thing as a triangular pyramid.
yes there is such a thing
Yes, it is called a tetrahedron or a triangular pyramid.