One side of a triangle has length 8 units and a second side has length 5 units. Which of the following values could be the area of the triangle?

Introduction / Context:
This conceptual question checks a candidate understanding of how the area of a triangle depends on the included angle between two sides. Given two fixed side lengths, the maximum possible area occurs when the angle between them is 90 degrees. This type of problem appears in aptitude tests to test reasoning about possible and impossible values, not just direct calculations.

Given Data / Assumptions:
• Two sides of a triangle have lengths 8 units and 5 units.• The third side and the included angle are not specified.• We must determine which listed area values could occur for some configuration of the triangle.

Concept / Approach:
For a triangle with sides a and b and included angle θ, the area A is given by A = (1/2) * a * b * sin(θ). For fixed a and b, the sine function ranges from 0 to 1, so the area ranges from 0 to a maximum of (1/2) * a * b when sin(θ) = 1. Here, a = 8 and b = 5, so maximum area is (1/2) * 8 * 5 = 20 square units. Therefore, any possible area must be less than or equal to 20 and greater than 0. Among the given options, only 20 satisfies this requirement.

Step-by-Step Solution:
Step 1: Use the area formula with two sides and included angle.A = (1/2) * a * b * sin(θ).Step 2: Substitute a = 8 and b = 5.A = (1/2) * 8 * 5 * sin(θ).Step 3: Simplify the constant part.(1/2) * 8 * 5 = 20.So A = 20 * sin(θ).Step 4: Note the range of sin(θ).0 < sin(θ) ≤ 1 for a non degenerate triangle.Step 5: Conclude the possible area range.0 < A ≤ 20.Step 6: From the options 24, 20, 26, 34, only 20 falls in this range.

Verification / Alternative check:
We can verify that the area 20 actually occurs when θ = 90 degrees. In that case the triangle is right angled with legs 8 and 5, and the area formula for a right triangle is A = (1/2) * base * height = (1/2) * 8 * 5 = 20. Thus 20 is not just theoretically possible but exactly realised by a right triangle with these sides as legs.

Why Other Options Are Wrong:
Option A: 24 is greater than the maximum area 20 and so cannot occur.Option C: 26 is even larger and is impossible for any angle between sides 8 and 5.Option D: 34 greatly exceeds the maximum area and is clearly impossible.

Common Pitfalls:
Some learners incorrectly assume that any positive area is possible as long as the triangle inequality holds, without considering the maximum area for given sides. Others may attempt to use Heron formula without knowing the third side, which is not practical here. Remembering that the largest area for fixed sides is obtained with a right angle, and computing that maximum value, provides a quick and reliable way to answer these questions.

Final Answer:
The area that could be the area of the triangle is 20 square units.