Devante is proving that perpendicular lines have slopes that are opposite reciprocals. he draws line m and labels two points on the line as (a, 0) and (0, b) . enter the answers, in simplest form, in the boxes to complete the proof. the slope of line m is . rotate line m 90° clockwise about the origin to get line n. the labeled points on line m map to (0, −a) and (, ) on line n. the slope of line n is . the slopes of the lines are opposite reciprocals because the product of the slopes is .

The slope of line m is (-b/a).

The labelled points on line m map to (0,-a) and (b,0) on the line n.

The slope of line n is (a/b).

The slopes of the lines are opposite reciprocals because the product of the slopes is -1.

Step-by-step explanation:

The slope of a line connecting the points (x₁,y₁) and (x₂,y₂) is

The slope of line m is

The line is rotated clockwise by an angle 90° to get line n.

The coordinates of the line n are (0,-a) and (b,0)

The slope of line n is

We see that,

m₁m₂ = -(b/a) * (a/b) = -1

Hence the products of the slopes of perpendicular lines is -1.

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!