To determine if an isosceles triangle is a right triangle, we need to use the properties of both isosceles and right triangles.
An isosceles triangle has two sides that are of equal length, which we'll call [tex]\(a\)[/tex]. The longest side of the triangle is called the hypotenuse, denoted by [tex]\(c\)[/tex].
In a right triangle, the Pythagorean theorem must be satisfied. The Pythagorean theorem states that for a right triangle with legs of lengths [tex]\(a\)[/tex] and [tex]\(b\)[/tex], and hypotenuse of length [tex]\(c\)[/tex], the following equation holds:
[tex]\[a^2 + b^2 = c^2\][/tex]
Since this is an isosceles right triangle, the legs [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are equal. Thus, we can set [tex]\(a = b\)[/tex].
Substituting [tex]\(a\)[/tex] for [tex]\(b\)[/tex] in the Pythagorean theorem, we get:
[tex]\[a^2 + a^2 = c^2\][/tex]
Combining like terms on the left side:
[tex]\[2a^2 = c^2\][/tex]
Thus, the equation that can be used to determine if an isosceles triangle is a right triangle is:
[tex]\[2a^2 = c^2\][/tex]
Therefore, the correct answer is:
[tex]\[2a^2 = c^2\][/tex]
