The question doesn't tell us if this is a right triangle, so we can't assume that it is. But there is a formula to find the area when we don't know the height: area = [p(p – a)(p – b)(p – c)]^1/2, where a, b, and c are the side lengths and p is half of the perimeter. The perimeter is 18 + 24 + 30 = 72, so p = 72/2 = 36.

Then area = [36(36 – 18)(36 – 24)(36 – 30)]^1/2 = [36 * 12 * 6 * 18]^1/2 = 216.

Since we are told nothing about the angles we cannot assume that these are isosceles triangles and are open to possibilites such as that shown below in which the left side would be larger. If both were isosceles triangles then the right side would be larger.

This question draws from the Third Side Rule of triangles. The length of any side of a triangle must be greater than the difference between the other sides and less than the sum of the other two sides.

This means that side x must be between 2 and 8 since the difference between 5 – 3 = 2 and the sum of 3 + 5 = 8.

Choices 3, 4, and 6 all fall within the range of 2 to 8, but choice 9 does not. The answer is 9.

Two sides of a triangle must add up to greater than the third side. 25, 37, 66 cannot be the lengths of the sides of a triangle as 25 + 37 < 66. 

If two sides of a triangle are 6 and 12, the third must be greater than 12-6 and less than 12+6 since two sides cannot be summed to be greater than the third side in a triangle.  There are 11 possible values for x: 7, 8, 9, 10,11, 12, 13, 14, 15, 16, 17. 

12: The sum of two sides of a triangle must be greater than the third side. Therefore, the length of the third side would have to be less than 12 and greater than 2. 

For a triangle where the length of two sides,  and , is the only information known, the third side, , is limited in the following matter:

For the triangle given:

.

Both choices A and B satisfy this criteria.

If two sides of a triangle are known and all angles are unknown, the length of the third side is limited by the difference and sum of the other two sides.

The missing side must be greater than .

Quantity A is greater.

Seeing the sides  and  may bring to mind a  triangle. However, we've been told nothing about the angles of the triangle. It could be right, or it could be obtuse or acute.

Since the angles are unknown, the side is bounded as follows:

There are plenty of potential lengths that fall above and below . The relationship cannot be determined.

If two sides of a triangle are known and the angles of the triangle are unknown, the length of the missing side is limited by the difference and sum of the other two sides.

For a triangle with sides  and , there is no way a side could be .

Quantity B is greater.