Acoordinate grid is used to make a map of the park. the location of the drinking fountain is (-4, 2); the location of the bench is (0, 2); and the location of the center of the slide is (0, 5).
(a) what is the distance from the drinking fountain to the bench? what is the distance from the bench to the center of the slide? explain your reasoning.
(b) what type of triangle is created by the three locations? explain.

4; 3; right

Step-by-step explanation:

The two points that define the segment from the drinking fountain to the bench are (-4, 2) and (0, 2).  This is a horizontal segment.  The distance of this segment is |-4-0| = |-4| = 4.

The two points that define the segment from the bench to the center of the slide are (0, 2) and (0, 5).  This is a vertical segment.  The distance of this segment is |5-2| = |3| = 3.

The slope of a horizontal segment is 0.  The slope of a vertical segment is undefined; this is obtained by dividing by 0.  This means these slopes are negative reciprocals, which is the definition of perpendicular line segments.  This means this is a right triangle.

step-by-step explanation:

first, eliminate the parenthesis:

3

x

+

2

−

4

x

second, group like terms:

3

x

−

4

x

+

2

third, combine like terms:

(

3

−

4

)

x

+

2

→

−

1

x

+

2

→

−

x

+

2

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!