Similar triangles are proportional.

Base₁ / Height₁ = Base₂ / Height₂

6 / 9 = 20 / Height₂

Cross multiply  and solve for Height₂

6 / 9 = 20 / Height₂

6 * Height₂=  20 * 9

Height₂=  30

Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.

Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.

We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.

We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:

 a² + b² = c²

15² + 20² = c²

625 = c²

c = 25

The length of BD is 25.

We now have what we need to find the perimeter of the quadrilateral.

Perimeter = sum of the lengths of AB, BC, CD, and DA.

Perimeter = 20 + 18.75 + 31.25 + 15 = 85

The answer is 85. 

The points define a 3-4-5 right triangle.  Its area is A = 1/2bh = ½(3)(4) = 6.  The scale factor (SF) of the new triangle is 3.  The area of the new triangle is given by A_new = (SF)² x (A_old) =

3² x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).

NOTE:  For a volume problem:  V_new = (SF)³ x (V_old).

We will use the Pythagorean Theorem to solve for the missing side length.

Since the perimeter of the ring is  feet and the ring is a square, solve for the length of a single side of the ring by dividing by .

The distance between the two boxers in opposing corners is a straight line from any one corner to the other. That straight line forms the hypotenuse of a right triangle whose other two sides are each  feet long (since they are each the sides of the square). 

Solving for the length of the hypotenuse of this right triangle with the pythagorean theorem  provides the distance between the two boxers when they are in opposite corners.

Using Pythagorean Theorem, we can solve for the length of leg x:

x² + 6² = 10²

Now we solve for x:

x² + 36 = 100

x² = 100 – 36

x² = 64

x = 8

Also note that this is proportionally a 3/4/5 right triangle, which is very common. Always look out for a side-to-hypoteneuse ratio of 3/5 or 4/5, or a side-to-side ratio of 3/4, in any right triangle, so that you may solve such triangles rapidly.

Using the pythagorean theorem, 8²=7²+x². Solving for x yields the square root of 15, which is 3.9

Using Pythagorean Theorem, we can solve for the length of leg x:

x² + 2² = (√8)² = 8

Now we solve for x:

x² + 4 = 8

x² = 8 – 4

x² = 4

x = 2

Use the Pythagorean Theorem. The sum of both legs squared equals the hypotenuse squared. 

Use the Pythagorean theorem: .

We know the length of one side and the hypotenuse.

Now we can solve for the missing side.