In a certain triangle that has an area of 12 square inches, the length (September 2023 - G01)

Correct answer: C. 6.

Let's solve this problem to find the base length of the triangle:

Area Formula: We know the area of the triangle is 12 square inches. The area of a triangle can be calculated using the formula: Area = 1/2 * base * height (altitude).
Height and Base Relationship: We are given that the height (altitude) is 2/3 the length of the base we want to find. Let B represent the base length. Then, the height (altitude) is H = 2/3 * B.

Substitute into Area Formula: Now, we can substitute the expressions for base and height into the area formula:
12 sq in = 1/2 * B * (2/3 * B)
Solve for Base (B):
Simplify the equation:
12 sq in = (1/6) * B^2
Multiply both sides by 6 to isolate B^2:
B² = 72 sq in

Take the square root of both sides to find B (be aware that squaring a negative number also results in a positive answer, so consider both positive and negative square roots):
B = ±√72 sq in

Since the base length cannot be negative, we discard the negative solution.
B = √72 sq in = 6√2 in (approximately 8.49 in)

Consider answer choices: The answer choices only have integer values. Since the exact answer is irrational (with a square root), a reasonable approximation would be the closest integer value.
Therefore, the answer is C. 6 inches (which is the closest integer value to the approximate answer of 8.49 inches).