Quantity A: area = base * height / 2 = 10 * 24 / 2 = 120

Quantity B: 2 * area = 2 * base * height / 2 = base * height = 5 * 12 = 60

Therefore Quantity A is greater.

Quantity A: Area = 1/2 * b * h = 1/2 * 6 * 7 = 42/2 = 21

Quantity B: Area = 1/2 * (b₁ + b₂) * h = 1/2 * (6 + 10) * 6 = 48

                Half of the area = 48/2 = 24

Quantity B is greater.

This is easier to see when the triangle is divided into six parts (blue). Each one contains an angle which is half of 120 degrees and contains a 90 degree angle. This means each triangle is a 30/60/90 triangle with its long side equal to the radius of the circle. Knowing that means that the height of each triangle is  and the base is .

Applying  and multiplying by 6 gives ). 

A triangle with a fixed perimeter does not have to have a fixed area. For example, a triangle with sides 3, 4, and 5 has a perimeter of 12 and an area of 6. A triangle with sides 4, 4, and 4 also has a perimeter of 12 but not an area of 6. Thus the answer cannot be determined.

Draw the triangles

Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse

Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is  times the side opposite the 30 degree angle. Thus, , which is about 8.66. This is larger than 7.5. 

The ratio of the sides of a 30-60-90 triangle is , with the hypotenuse being . Thus, the perimeter of this triangle would be . Since the triangle depicted in this problem has a perimeter of ,  must equal 1, which would make the hypotenuse equal to 2.

Use the Pythagorean theorem,   , with   and  , and solve for .

Rearrange to isolate :

Use FOIL to multiply out:

Distribute the minus sign to rewrite without parentheses:

Combine like terms:

Take the square root of both sides:

At first sight, it's tempting to assume this is a right triangle and to thus use the Pythagorean Theorem to find a length of 5 for the missing side.

However, the triangle was not stated to be a right triangle in the problem statement, and no indication was given in the drawing to indicate that it was a right triangle either, such as a square demarcation in the vertex opposite the side measuring 13.

Thus there is not enough information to give the length of the missing side. When taking standardized math tests, be careful making assumptions about information that is not given.

If an equilateral triangle is divided in 2, it forms two 30-60-90 triangles. Therefore, the side of the equilateral triangle is the same as the hypotenuse of a 30-60-90 triangle. The side lengths of a 30-60-90 triangle adhere to the ratio x: x√3 :2x. since we know the hypothesis is 4, we also know that the base is 2 and the height is  2√3.