This problem requires a bit of creative thinking (unless you have memorized the fact that an equilateral triangle always has an area equal to its side length times .

Consider the equilateral triangle:

Since this kind of triangle is a species of isoceles triangle, we know that we can drop down a height from the top vertex. This will create two equivalent triangles, one of which will look like:

This gives us a 30-60-90 triangle. We know that for such a triangle, the ratio of the side across from the 30-degree angle to the side across from the 60-degree angle is:

We can also say, given our figure, that the following equivalence must hold:

Solving for , we get:

Now, since , we know that  must be smaller than . This means that  or . Quantity B is larger than quantity A.

If the perimeter of our equilateral triangle is , each of its sides must be  or . This gives us the following figure:

Since this kind of triangle is a species of isoceles triangle, we know that we can drop down a height from the top vertex. This will create two equivalent triangles, one of which will look like:

This gives us a 30-60-90 triangle. We know that for such a triangle, the ratio of the side across from the 30-degree angle to the side across from the 60-degree angle is:

Therefore, we can also say, given our figure, that the following equivalence must hold:

Solving for , we get:

Now, since , we know that  must be smaller than . This means that  or 

Therefore, quantity B is larger than quantity A.

The internal angles of a triangle must add up to 180. 180 - 75 -60= 45. 

To find the perimeter, add up the sides, here 5 + 12 + 13 = 30. To find the area, multiply the two legs together and divide by 2, here (5 * 12)/2 = 30.

If B is a midpoint of AC, then we know AB is 12. Moreover, triangles ACE and ABD share angle DAB and have right angles which makes them similar triangles. Thus, their sides will all be proportional, and BD is 4. ¹/₂bh gives us ¹/₂ * 12 * 4, or 24.

Area = 1/2(base)(height). You could use Pythagorean theorem to find the height or, if you know the special right triangles, recognize the 5-12-13. The area = 1/2(12)(5) = 30. 

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.