Expert: Sherman D. - 10/3/2005

Question
Hello, I received a question on my test and I did not know how to approach the question. My teacher would like us to re-do the questions that we got wrong. Here's the one that I need help in:

City A is 30 miles directly north of City C, and City B is 40 miles directly east of City C. Cities A and B are connected by a straight road. Find the length of the shortest path from City C to the road that connects A and B.

Hint: The shortest path, segment CD, is the length of the perpendicular from C to segment AB. First, find AB, AD, and BD.



I would really appreciate if you will be able to take out the time to help me. Thank you in advance!

Answer
Once i knew BC and AC, i just use the pythagorean thereom, which gave me 50 miles.

when you create CD, you have length of BD and AD

since we knew BD, you can use the pythagoream thereom.

(BD)^2 + (CD)^2 = (BC)^2
(BD)^2 + (CD)^2 = 40^2
(BD)^2 + (CD)^2 = 1600

and

(AD)^2 + (CD)^2 = (AC)^2
(AD)^2 + (CD)^2 = 30^2
(AD)^2 + (CD)^2 = 900

since CD for AC,AD and CB,BD is the same length, we would just work it like this

(BD)^2 + (CD)^2 = 1600
(CD)^2 = -(BD)^2 + 1600

and

(AD)^2 + (CD)^2 = 900
(CD)^2 = -(AD)^2 + 900

Now just set the 2 problems equal to each other like this

-(BD)^2 + 1600 = -(AD)^2 + 900
-((BD)^2 - 1600) = -((AD)^2 - 900)

the negative values cancel out

(BD)^2 - 1600 = (AD)^2 - 900

add 1600 to both sides

(BD)^2 = (AD)^2 + 700

sqrt both sides

BD = sqrt((AD)^2 + 700)

since we know that BD is AD - 50, we can work the problem like this

AD - 50 = sqrt((AD)^2 + 700)

square both sides

(AD - 50)^2 = (AD)^2 + 700
(AD - 50)(AD - 50) = (AD)^2 + 700
(AD)^2 - 50(AD) - 50(AD) + 2500 = (AD)^2 + 700
(AD)^2 - 100(AD) + 2500 = (AD)^2 + 700

the (AD)^2s cancel out

-100(AD) + 2500 = 700

-100(AD) = -1800

the negatives cancel out

AD = 18

BD = -(18) + 50
BD = -18 + 50
BD = 32

now just use

(CD)^2 = -(AD)^2 + 900
(CD)^2 = -(18)^2 + 900
(CD)^2 = -324 + 900
(CD)^2 = 576

sqrt both sides

CD = 24

Try this with (CD)^2 = -(BD)^2 + 1600 and you will get the same answer.

so CD = 24 miles

Was that one of the choices.