Question

a graphic designer is creating a logo for a client. Lines DB and AC are perpendicular.The equation of DB is 1/2x+2y=12 what is the equation of AC

Answer 1

Answer: -4x + y= -28

Explanation:

12⁢x+2⁢y=12y=-14⁢x+6The slope of line DB is -14. The slopes of perpendicular lines are opposite reciprocals of each other, so line AC will have a slope of 4. Substitute the slope of 4 and the point (8,4) into the slope-intercept formula to find the equation of line AC.4=4⁢(8)+b4=32+b-28=bSo, line A⁢C has a y-intercept, b, of -28 and a slope of 4. Convert the equation to standard form.

Answer 2

y=4x-28

Explanation

when two lines are perpendicular the product o their slopes equals, -1

`\begin{gathered} if \\ line_1\perp\text{ line}_2 \\ then \\ slope_1*slope_2=-1 \end{gathered}`

so

Step 1

find the slope of the line AC

a)the function of the line DB is given in the form

`\begin{gathered} y=mx+b \\ where \\ m\text{ is the slope} \\ b\text{ is the y-intercept} \end{gathered}`

so

`(1)/(2)x+2y=12`

b)isolate y to get the slope-intercept form, and check the m value

`\begin{gathered} (1)/(2)x+2y=12 \\ subtract\text{ }(1)/(2)x\text{ in both sides} \\ (1)/(2)x+2y-(1)/(2)x=12-(1)/(2)x \\ 2y=-(1)/(2)x+12 \\ divide\text{ both sides by 2} \\ (2y)/(2)=(-(1)/(2)x+12)/(2) \\ y=-(1)/(4)x+6\Rightarrow line\text{ 1} \\ so \\ slope_1=-(1)/(4) \end{gathered}`

c) now, as the lines AC and DB are perpendicular replace to find slope 2

`\begin{gathered} slope_1*slope_2=-1 \\ -(1)/(4)*slope_2=-1 \\ Multiply\text{ both sides by -4} \\ -(1)/(4)slope_2*-4=-1*-4 \\ slope_2=4 \end{gathered}`

therefore

the slope of the line AC is 4

Step 2

now, we can use the slope-point formula to find the equaiton of the line,it says

`\begin{gathered} y-y_1=m(x-x_1) \\ where\text{ m is the slope} \\ (x_1,y_1)\text{ is a point from the line} \end{gathered}`

so

a)let

`\begin{gathered} slope=4 \\ point=(8,4)\text{ this is a common point for both lines} \end{gathered}`

b) finally, replace and solve for y

`\begin{gathered} y-y_(1)=m(x-x_(1)) \\ y-4=4(x-8) \\ y-4=4x-32 \\ add\text{ 4 in both sides} \\ y-4+4=4x-32+4 \\ y=4x-28 \end{gathered}`

so, the equation of the line AC ia

y=4x-28

I hope this helps you