For a triangle, the sum of the two shortest sides must be greater than that of the longest.  We are given two sides as 5 and 12.  Our third side must be greater than 7, since if it were smaller than that we would have  where is the unknown side.  It must also be smaller than 17 since were it larger, we would have .

Thus our perimeter will be between and . Only II and III are in this range.

To calculate the perimeter, we simply need to add the three sides of the triangle.

Therefore, the perimeter is , which is the final answer.

Since BD is the height of triangle ABC, we can apply the Pythagorean Theorem to, let's say, triangle DBC and .

Since the basis of the height is at the midpoint of AC, it follows that triangle ABC, is an isoceles triangle.

We can find the perimeter by multiplying BC by 2 and add the basis of the triangle AC, which has length of .

The final answer is therefore .

For any given triangle, the perimeter  is the sum of the lengths of its sides. Given side lengths of , , and , . 

For any given triangle, the perimeter  is the sum of the lengths of its sides. Given side lengths of , , and , . 

For any given triangle, the perimeter  is the sum of the lengths of its sides. Given side lengths of , , and , . 

If you know either that  or , you have three congruencies - two sides and a nonincluded angle. This is not enough to establish triangle congruence,

If you know that , this, along with the other two statements, establishes that ; all this proves is that both triangles are isosceles.

If you also know that , however, the three statements together make three side congruencies, setting up the Side-Side-Side criterion for triangle congruence.

We use the congruence of corresponding sides and angles of congruent triangles to prove four of these statements:

, so  bisects .

, so  is the midpoint of .

, and, since they form a linear pair, they are supplementary - therefore, they are right angles. Also, by definition, this makes  an altitude.

However, nothing in this congruence proves that  is congruent to the other two sides of  (which are congruent). The correct statement to exclude is that  is equilateral.

An isosceles triangle has two congruent angles by the Isosceles Triangle Theorem; these are the base angles. Since at least two angles of any triangle must be acute, a base angle must be acute - that is, it must measure under . The only choice that does not fit this criterion is , making this the correct choice.

If these are the measures of the interior angles of a triangle, then they total . Add the expressions, and solve for .

One angle measures . The others measure:

All three angles measure less than  , so the triangle is acute. Also, there are two congruent angles, so by the converse of the Isosceles Triangle Theorem, two sides are congruent, and the triangle is isosceles.