Two sides of a triangle have length 6 and 8. Which of the following ar

MAGOOSH OFFICIAL SOLUTION:

This requires a picture explanation. Consider the leg of 6 and the leg of 8 attached at a "hinged" joint at B.

Here, the angle is made very narrow, only 1/100 of a degree. The area of this triangle would be 0.00419 --- you don't need to be able to calculate something like this for the test. The point is -- if the angle were one millionth, or one billionth, the area could be really really small --- greater than zero, but a really tiny decimal. So, clearly, the area can be less than two.

Here, I moved the legs of 6 and 8 apart a little, and produced a triangle with an area of exactly two:

Again, you don't have to know how to build a triangle like this: this is just to show you that it is, indeed possible.

As we increase the angle, we get triangles with more and more area. You do not need to know how to find these areas --- this is just to demonstrate that the area would increase.



Now, there's a geometry theorem that states, when the angle is 90°, the area of the triangle will be a maximum. At that point, the length of 8 will be the base, and the length of 6 will be the height, so the area = 0.5*bh = 24, the maximum possible area.

Thus, the triangle can have all three areas: I, II, and III.

Answer = E.