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

Before we can set up a proportion, we need to know the third side of the larger triangle, since that side is actually the one that corresponds to 6, since both are the shortest sides in their respecitve triangles. We can solve for that side using the Pythagorean Theorem,

subtract 100 from both sides

take the square root

Now we can set up a proportion comparing corresponding sides to solve for x:

 cross-multiply

divide by 7.5

The Law of Cosines is as follows:

Notice these equations contains the Pythagorean Theorem, , within it.

The term at the end is the adjusting term for triangles which are not right triangles. 

This is a prime example of a case that calls for using the Law of Cosines, which states

where , , and  are the three sides of the triangle, and  is the angle opposite side .  Looking at our triangle, taking , then we have , , and .  Plugging this into our formula, we get.

Using our calculator to approximate the cosine value gives

Simplifying further gives 

Solving by taking the square root gives

To apply the Law of Cosines,  is the unknown,  and  are the respective given sides, and the given angle is .

Therefore, the equation becomes: 

Which yields 

Add  to the other two given sides to get the perimeter,