Question

Find the distance from the given point to the given line. Estimate your answer to the nearest 2 decimal places.

Answer 1

did u need to find the shortest distance?

Answer 2

yes, distance means shortest distance

Answer 3

Can someone break this down for me?

Answer 4

it will take 3 steps
1) find the equation of the line perpendicular to \(y=-2x-8\) through the point \((-2,6)\)
2) find the intersection of the two lines
3) find the distance between the point of intersection and the point \99-2,6)\)

be happy to help with any of the steps

Answer 5

Thank you! could you take me through them?

Answer 6

sure
for the first one, the line \(y=-2x-8\) has slope \(-2\) so the perpendicular line will have slope \(\frac{1}{2}\)

the point slope formula gives
\[y-6=\frac{1}{2}(x+2)\]
\[y-6=\frac{1}{2}x+1\]
\[y=\frac{1}{2}x+7\]

Answer 7

for the second part, set
\[\frac{1}{2}x+7=-2x-8\] and solve for \(x\)
\[x+14=-4x-16\]
\[5x=-30\]
\[x=-6\]
since \(x=-6\) we get \(y=-2\times -6-8=12-8=4\) so the point of intersection is \((-6,4)\)

Answer 8

finally use the distance formula to find the distance between the points \((-6,4)\) and \((-2,6)\)

Answer 9

\[\sqrt{(-6+2)^2+(6-4)^2}=\sqrt{4^2+2^2}=\sqrt{20}\]

Answer 10

you got this?

Answer 11

I think so

Answer 12

k
if you have a question about any step, ask

Answer 13

can you do one more for me?

Answer 14

why not

Answer 15

ok thank you! and so for the first one it would be 20?

Answer 16

no it is not 20, it is the square root of 20 i.e.
\[\sqrt{20}\] or
\[2\sqrt{5}\] they are the same number

Answer 17

so how would I break it down?

Answer 18

not sure what you mean
the distance formula gives the answer as \(\sqrt{20}\) and since \(2=4\times 5\) we can write
\[\sqrt{20}=\sqrt{4\times 5}=\sqrt{4}\sqrt{5}=2\sqrt{5}\]

Answer 19

i mean since \(20=4\times 5\)

Answer 20

if you want a decimal approximation, you need to use a calculator

Answer 21

(-2, 6), y = -2x - 8

Answer 22

this is what you wrote above right? find the distance between the point and the line

Answer 23

yes I needed help

Answer 24

i wrote all the steps above to find the distance. the final answer is \(\sqrt{20}\)

Answer 25

no I mean the other lol

Answer 26

other problem

Answer 27

what is the other problem?

Answer 28

(-2, 6), y = -2x - 8

Answer 29

i think you wrote the same one down twice

Answer 30

ohhh sorry this one

Answer 31

(-6,4), y = -2x + 7

Answer 32

same idea
equation of the perpenicular line is
\[y-4=\frac{1}{2}(x+6)\] \[y-4=\frac{1}{2}x+3\]\[y=\frac{1}{2}x+7\]

Answer 33

set the two lines equal, get
\[\frac{1}{2}x+7=-2x+7\] so \(x=0\) and therefore \(y=7\) for the point of intersection

Answer 34

then find the distance between \((0,7)\) and \((-2,6)\) you get
\[\sqrt{2^2+(7-6)^2}=\sqrt{5}\]

Answer 35

thank you!

Answer 36

yw,
happy to help a tv star!