Given the context of the Pythagorean theorem and the areas associated with the triangle \( \triangle ADG \) with sides of lengths 3, 4, and 5 units, we can analyze the statements provided in relation to the areas formed by the squares constructed on the triangle's sides.
The triangle has:
- Side \( AD = 3 \) units, so the area of the square on \( AD \) is \( 3^2 = 9 \) square units.
- Side \( DG = 4 \) units, so the area of the square on \( DG \) is \( 4^2 = 16 \) square units.
- Hypotenuse \( AG = 5 \) units, so the area of the square on \( AG \) is \( 5^2 = 25 \) square units.
According to the Pythagorean theorem: \[ AD^2 + DG^2 = AG^2 \] which translates into: \[ 9 + 16 = 25 \] This equation confirms the relationship \( \text{Area of square on } AD + \text{Area of square on } DG = \text{Area of square on } AG \).
Now, let's evaluate the statements based on these area calculations:
- A: Area(ABCD) - AREA(DGA) = AREA(DEFG)
- B: Area(ABCD) - AREA(GHIA) = AREA(DGA)
- C: Area(ABCD) + AREA(DGA) = AREA(GHIA)
- D: Area(ABCD) + AREA(GHIA) = AREA(DEFG)
- E: Area(DEFG) + AREA(GHIA) = AREA(ABCD)
From what we established:
- The area of the square on \( AD \) (9) and the area of the square on \( DG \) (16) summing to the area of the square on \( AG \) (25) can imply specific relationships regarding the areas if we consider comparable areas.
Given that:
- C corresponds directly to the Pythagorean theorem \( \text{Area of square on } AD + \text{Area of square on } DG = \text{Area of square on } AG \), which can be denoted as \( 9 + 16 = 25 \).
Based on the above analysis, statement C: \( \text{Area(ABCD)} + \text{AREA(DGA)} = \text{AREA(GHIA)} \) is true as it aligns directly with the Pythagorean theorem since we verify the relationship between the areas related to the three sides.
Thus, the correct answer is: C: Area(ABCD) + AREA(DGA) = AREA(GHIA)