If the smallest angle of a triangle is 20° and it is included between sides of 4 and 7, then (to the nearest tenth) the smallest side of the triangle is .

Step-by-step explanation:

The smallest side of a triangle is formed by the smallest angle in the triangle.

To find the side opposite (formed by) the 20 degree angle, we can use the Law of Cosines. The Law of Cosines states that for any triangle, , where , , and are the three sides of the triangle and is the angle opposite to .

Let be the side opposite to the 20 degree angle.

Assign variables:

Substituting these variables, we get:

Therefore, the shortest side of this triangle is 3.5.

a reflection across the x axis and a 90 degree rotation left and right.

there is only one way to arrange all the eight relative to each other because they are not distinct. we have

now there are nine places to place an (and not more than one in one place).   because we have four symbols, we need to choose four of these nine places to insert an . order does not matter because the are not distinct.

thus there are 1 · c(9,4) ways, or, if we calculate it,

thus there are 126 ways to form such "words."