4. Supplementary angles together make a straight line (two right angles). So BOE is supplementary to BOC
Answer: ∠BOE
5. Distance and midpoint between (4,-2), (6,8)

midpoint

Answer: distance 2√26, midpoint (5, 3)
6.That's a rectangle oriented parallel to the axes, width parallel to the x axis of 7 - -3 = 10 and length along y of 1 - - 4 = 5, so an area of 10(5)=50
Answer: 50
The midpoint formula for a segment is:

apply to points R and P

using the definition of slope find the slope of the segment

apply to points R and P

to lines are parallel when the slopes are the same

two lines are perpendicular when the product of the slopes is equal to -1
Answer:

Step-by-step explanation:

This is a homogeneous linear equation. So, assume a solution will be proportional to:

Now, substitute
into the differential equation:

Using the characteristic equation:

Factor out 

Where:

Therefore the zeros must come from the polynomial:

Solving for
:

These roots give the next solutions:

Where
and
are arbitrary constants. Now, the general solution is the sum of the previous solutions:

Using Euler's identity:


Redefine:

Since these are arbitrary constants

Now, let's find its derivative in order to find
and 

Evaluating
:

Evaluating
:

Finally, the solution is given by:

(1) The train travels 4 miles per gallon.
(2) The slope of the graph is 
Explanation:
(1) The miles that train travels per gallon is given by

Dividing, we have,

Thus, the train travels 4 miles per gallon.
(2) To determine the slope, let us consider two points from the graph.
The coordinates are
and 
Thus, substituting the coordinates in the slope formula, we get,

Simplifying, we have,

Thus, the slope of the graph is 
Answer:
3.18, 3.21, 3.23
Step-by-step explanation:
You just nee to find some decimals that round up to 3.2