Find the coordinates of the circumcenter of def with coordinates d(1,3) e(8,3) and f(1,-5). show your work

As shown in attached figure C is the circumcenter of triangle DEF

First find midpoint of DF using mid point formula

----------------------------(1)

for two points and

so mid point M of D(1,3) and F(1,-5) will be

M : (1,-1)

Find slope of DF using slope formula

-----------------------------------------(2)

So slope of D(1,3) and F(1,-5) will be will be

Now line CM is the perpendicular bisector of line DF. So slope of line CM will be negative reciprocal of slope of DF

So slope of CM will be or simply 0

So now we will write equation of line CM using slope m =0 and passing through mid point M (1,-1) using formula

on plugging values we get

y + 1 =0

y +1 -1 = 0 -1

y = -1 ---------------------------------------------------(3)

So thats the equation of line CM

Similarly now line midpoint N of line DE using mid point formula from (1)

so mid point N of D(1,3) and E(8,3) will be

N : (4.5,3)

Find slope of DE using slope formula in (2)

So slope of D(1,3)) and F(8,3) will be will be

or simply 0

Now line CN is the perpendicular bisector of line DE. So slope of line CN will be negative reciprocal of slope of DE

So slope of CN will be be reciprocal of 0 which is undefined when 0 goes in denominator. All undefined slope lines have equation of form for point on it

So for line CN with undefined slope and passing through point (4.5,3) its equation will be

x = 4.5 ----------------------------------------------------------------------------(4)

Solving equations (3) and (4) we get coordinates of intersection point C which will be

x=4.5, y =-1

so answer for circumcenter C coordinates will be (4.5,-1)

step-by-step explanation:

from the graph attached below:

domain of graph is the value of x for which the given function profit function p(x) is defined.

(a) the domain represents the number of years for which the outlet will earn a profit.

range of profit function defines the lowest to highest value of profit earned.

(b) the range of this function is (0 , 160 ). for this function, the range represents the outlet's profit earned.

answer: learning objective(s)

·         describe the creation and use of systems of equations.

·         graph a system of linear equations on the coordinate plane and identify its solution.

introduction

sometimes graphing a single linear equation is all it takes to solve a mathematical problem. other times, one line just doesn’t do it, and a second equation is needed to model the situation. this is often the case when a problem involves two variables. solving these kinds of problems requires working with a system of equations, which is a set of two or more equations containing the same unknowns.

let’s take a look at systems of equations, and see what the graphs of individual equations within a system reveal about the mathematical relationship of the variables.

systems of equations

a system of equations contains two or more linear equations that share two or more unknowns. to find a solution for a system of equations, we must find a value (or range of values) that is true for all equations in the system.

the graphs of equations within a system can tell us how many solutions exist for that system. look at the images below. each show two lines that make up a system of equations (in the graph on the right the two lines are superimposed and look like a single line). how many points in common does each system of lines reveal?

one solution

no solutions

infinite solutions

if the graphs of the equations intersect, then there is one solution that is true for both equations.

if the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

if the graphs of the equations are the same, then there are an infinite number of solutions that are true for both equations.

remember, the graph of a line represents every point that is a possible solution for the equation of that line. so when the graphs of two equations cross, the point of intersection lies on both lines, meaning that it is a possible solution for both equations. when the graphs of two equations never touch, there are no shared points and there are no possible solutions for the system. when the graphs of two equations lie on top of one another, they share all their points and every one is a possible solution.

graphing as a solution method

graphing equations in order to identify a specific point of intersection is usually not a precise way to solve systems because it is often difficult to see exactly where two lines intersect (unless you are using a computer-based graphing program that allows you to zoom in on a point). however, the graph of a system of equations can still give a good idea of what type of solution, if any, exists. let’s graph a system, and see how it works.

graph the system y = 3x and x + 2y = 4. how many solutions does this system have?

a graph of the two lines y = 3x and x + 2y = 4 shows us that the lines intersect, meaning that there is a single (x, y) value that satisfies both equations. looking at the graph does not tell us exactly where that point is, but we don’t need to know that information here, because the question only asks for the number of solutions.

so a system made of two intersecting lines has one solution. now let’s look at a different situation:

how many solutions exist for the system y -0.5x = 7 and 2y = x − 3?

plotting both equations, it looks like there is no solution—the lines are parallel. to check this finding, we can compare the slopes of the equations. to make that easy, we’ll rearrange the equations into slope-intercept form, or y = mx + b.   that gives us the equations y = 0.5x + 7 and y = 0.5x − 1.5.   yes, the slope of both lines is 0.5, which means the lines really are parallel. they never intersect, so there is no point that lies on both lines, and no solution to the system.

step-by-step explanation: