Two triangles are similar if and only if their side lengths are proportional.

In this case, two of the sides are proportional, leading us to a scale factor of 2.

However, with the last side,  which is not our side length.

Thus, these pair of sides are not proportional and therefore our triangles cannot be similar. 

We must remember that there are three ways to prove triangles are similar.

1. At least two angles in one triangle are congruent to angles in another (AA)
2. All three pairs of corresponding sides are proportional (SSS)
3. Two pairs of corresponding sides are proportional and the angles between those sides are congruent (SAS)

Comparing triangles I and II, we only have one angle and two sides in trinagle II, so attempting to use either AA or SSS for similarity will not work, leaving SAS as the only option.  If we compare the two given sides in each triangle, we notice that the ratio of the longer side in triangle I to the longer side in triangle II is

The ratio of the shorter sides in each triangle are

Notice we have equal ratios and thus a proportion.  However, we still must confirm that the included angles are congruent.  The measure for this angle is not given in triangle I, but we can calculate since all three angles must add up to 180 degrees.  Calculation tells us that the measure is 98 degrees, which unfortunately does not equal the 110 from triangle II.  Therefore, we have no SAS and therefore no similarity between I and II.

Transitioning to I and III, we only have angles in triangle III, so we are unable to use either SSS or SAS.  However, we previously calculated the measure third angle in triangle I to be 98.  Therefore, two of our angles are congruent, meaning we have AA and thus similarity.

Regarding II and III, we can use some logic.  Since we know I and III are similar, then if II and III were also similar, then we could use the transitive property to conclude that I and II are also similar.  But we know this is false, so II and III cannot be similar.

Therefore, the only two similar triangles are I and III.

For both triangles, we are given the "legs." Based on their relative lenghts, we can see that 2 corresponds with 3, and 7 corresponds with 10.5. First we need to make sure that these two triangles are similar. We can do this by comparing the ratios of corresponding sides:

There are a couple of ways to go from here. One would be to cross-multiply:

the ratios are equal, so the triangles are similar, and the scale factor is .

Based on their positions relative to the congruent angles, and their relative lengths, we can see that 1.5 corresponds to 6, and 8 corresponds to 30. If the ratios of corresponding sides are equal, then the triangles are congruent: 

We can compare these in a couple different ways. One would be to cross-multiply:

These triangles are not similar.

To determine if the triangles are similar, set up a proportion.

2/4 = 4/8 = 5/10

 When we do this, we cross multiply to get a true statement.

Or, we can find the scale factor.

Since the scale factor is 2 for all three lengths, it becomes clear that these triangles are similar. 

The actual shift for  is .

Similar triangles by defnition have proportional sides. We can divide corresponding parts in this case to find the scale factor.

Corresponding parts are the two smallest sides, the medium sides, and the largest sides.

Thus: 

 is the scale factor. 

 Then, we use this to find the missing side. 

Therefore, .

We can find the scale factor by dividing 18 by 6.

We get that the scale factor is 3.

Multiplying the other side by 3, we get the new side, 21. 

In order to determine whether if the dimensions of the triangle are of the same proportions, the ratios of the dimensions must also be the same as the 3-4-5 triangle.

The following scale factors multiplied to the 3-4-5 triangle yield similar proportions.

The only dimensions that cannot be attained by multiplying a particular scale factor with the 3-4-5 is:

Initially we are given the hypotenuse and one leg of the large triangle, but both legs of the small triangle. To set up a proportion, we need to know both legs of the large triangle, and we can solve for the missing one using the pythagorean theorem:

, where a and b are the legs and c is the hypotenuse.

subtract 64 from both sides

take the square root of both sides

Now that we have both legs, we can see that 15 corresponds with x, and 8 corresponds with 6, so we can set up a proportion to solve for x:

divide both sides by 8