If the hypotenuse of the triangle is 6, what is the length of a side?

To determine why the choice of 3√2 is correct for the length of one side of a right triangle with a hypotenuse of 6, we can utilize the Pythagorean theorem.

In a right triangle, the relationship between the lengths of the sides is given by the equation:

( a^2 + b^2 = c^2 )

where ( c ) is the length of the hypotenuse, and ( a ) and ( b ) are the lengths of the other two sides. Assuming that the two sides are equal (which is a scenario for an isosceles right triangle), we can set ( a = b ).

Let’s denote the length of one side as ( a ). The equation then becomes:

( a^2 + a^2 = 6^2 )

This simplifies to:

( 2a^2 = 36 )

Dividing both sides by 2 gives:

( a^2 = 18 )

To find ( a ), we take the square root of both sides:

( a = \sqrt{18} )

which simplifies to:

( a = \sqrt{9 \times 2} = 3\sqrt