This problem requires us to use either the Law of Sines or the Law of Cosines. To figure out which one we should use, let's write down all the information we have in this format:

A = 35°       a = ?

B = ?          b = 8

C = ?          c = 12

Now we can easily see that we do not have any complete pairs (such as A and a, or B and b) but we do have one term from each pair (A, b, and c). This tells us that we should use the Law of Cosines. (We use the Law of Sines when we have one complete pair, such as B and b).

Law of Cosines:

Since we are solving for a, let's use the first version of the Law of Cosines.
This gives us:

This problem requires us to use either the Law of Sines or the Law of Cosines. To figure out which one we should use, let's write down all the information we have in this format:

A = 25°       a = 17

B = 50°       b = ?

C = ?          c = ?

Now we can easily see that we have a complete pair, A and a. This tells us that we can use the Law of Sines. (We use the Law of Cosines when we do not have a complete pair).

Law of Sines:

To solve for b, we can use the first two terms which gives us:

Use the pythagorean theorem:

The triangle has two sides of the same length, 18.4, so, by definition, it is isosceles.

Convert each of the three measures to the same unit; we will choose the smallest unit, millimeters.

One meter is equivalent to 1,000 millimeters, so 12 meters can be converted to millimeters by multiplying by 1,000:

One centimeter is equivalent to ten millimeters, so 1,200 cenitmeters can be converted to millimeters by multiplying by 10:

These two sides have the same length. However, the third side, which has length 12 millimeters, is of different length. Since the triangle has exactly two congruent sides, it is by definition isosceles, but not equilateral.

Convert each of the three measures to the same unit; we will choose the smallest unit, inches.

One foot is equal to twelve inches, so  feet can be converted to inches by multiplying by 12:

One yard is equal to 36 inches.

The lengths of the sides in inches are 18, 24, and 36. Since no two sides have the same measure, the triangle is by definition scalene.

The measures of the interior angles of a triangle add up to . If  is the measure of the third angle, then 

Solve for :

By the Isosceles Triangle Theorem, if two sides of a triangle have the same length, their opposite angles have the same measure. Since no two angles have the same measure, no two sides have the same length. This makes the triangle scalene.

The measures of the interior angles of a triangle add up to . If  is the measure of the third angle, then 

Solve for :

The triangle has two congruent angles - each with measure . As a consequence, by the Converse of the Isosceles Triangle Theorem, the triangle has two congruent sides, making it, by definition, isosceles.

The measures of the interior angles of a triangle add up to . If  is the measure of the third angle, then 

Solve for :

The triangle therefore has an angle with measure greater than  - an obtuse angle. The triangle is by definition an obtuse triangle,

First, convert all of the given sidelengths to the same unit; here, we choose the smallest unit, millimeters.

One meter is equal to 1,000 millimeters.

One centimeter is equal to 10 millimeters, so convert 250 centimeters to millimeters by multiplying by 10:

The measures of the sides of the triangle, in millimeters, are 1,000, 1,200, and 2,500.

By the Triangle Inequality, the sum of the measures of the shortest two sides of a triangle must exceed the length of the longest side, so for this triangle to be possible it must hold that

or

This is false, so a triangle with these sidelengths cannot exist.