What kind of triangle has the relations between a, b, c, sides: (3^1/2)*a - (3^1/2)*b = c a + b = (3^1/2)*c

We'll try to determine the length of b and the length of
c, considering the length of a side.

We'll multiply the
second relation with the value (3^1/2) and after that we'll add the equivalent obtained
relation to the first one.

(3^1/2)*a + (3^1/2)*b +
(3^1/2)*a - (3^1/2)*b = 3*c + c

We'll group the same
terms:

2*(3^1/2)*a =
4*c

(3^1/2)*a = 2
*c

c=
[(3^1/2)*a]/2

With the c value written in
function of "a" value, we'll go in the second relation and substitute
it:

a + b =
[(3^1/2)*(3^1/2)*a]/2

a + b =
3*a/2

We'll have the same denominator on the left side of
the equality:

2*a + 2*b =
3*a

2*b = 3*a - 2*a

2*b =
a

b =
a/2

If the triangle is a right one, then,
using the Pythagorean theorem, we'll have the following relation between the sides of
triangle:

a^2 = b^2 + c^2

Now,
we have to plug in the values of "b" and "c", in the relation
above:

a^2 = a^2/4 +
3*a^2/4

a^2 =
4*a^2/4

a^2 =
a^2

We've shown that the equality is true,
so the triangle is right, where "a" is hypotenuse and "b","c" are
cathetus.