Question

Locate a point C on the x-axis such that AC+BC is minimized. C=(_,0)

Answer 1

The endpoints of AB are A(0,2) and B(9,4)

Answer 2

check out this page

http://www.analyzemath.com/calculus/Problems/minimum_distance.html

Answer 3

I recommend you read over where it says "
METHOD 2: We "construct a virtual house" (projection) H on the other side of the river"

Answer 4

let me know if that helps or not

Answer 5

That just confuses me..

Answer 6

ok one sec

Answer 7

First plot the points A and B

Answer 8

If we use method 2, we reflect point B over the x axis to get point B'

Answer 9

Then you draw a straight line from A to B'

Answer 10

The line we just drew goes through (3,0)

so that is the point that minimizes the distance AC + BC

this is because the shortest distance between any two points is a straight line and because CB = CB' (you can prove this using CPCTC and the hypotenuse leg postulate)

Answer 11

Okay, that makes sense. is there an equation I can use to set it up to show my work?

Answer 12

nice

Answer 13

let C be the point (x,0)

the distance from A(0,2) to C(x,0) can be calculated using the distance formula

d = sqrt( (x1-x2)^2 + (y1-y2)^2 )

d = sqrt( (0-x)^2 + (2-0)^2 )

d = sqrt( x^2 + 4 )

so that is the distance from A to C

Answer 14

the distance from C to B is

d = sqrt( (x1-x2)^2 + (y1-y2)^2 )

d = sqrt( (x-9)^2 + (0-4)^2 )

d = sqrt( x^2 - 18x + 81 + 16 )

d = sqrt( x^2 - 18x + 97 )

Answer 15

The total distance AC + BC is the sum of those two distances

sqrt( x^2 + 4 ) + sqrt( x^2 - 18x + 97 )

you would then use calculus or a graphing calculator to find the min of this function to get the min distance

Answer 16

Okay, thanks so much for your help. Could you help me with one more problem?

Answer 17

sure, go ahead

Answer 18

Fill in the values to complete the reflection of AB with matrix multiplication.

Answer 19

That's all it says?

Answer 20

Yeah

Answer 21

well I found this page

http://www.mathplanet.com/education/geometry/transformations/transformation-using-matrices

but the problem is that I don't know what line we are reflecting over

Answer 22

are we reflecting over the x axis? y axis? or some other line?

Answer 23

x-axis

Answer 24

what are the coordinates of A and B?

Answer 25

same from that previous problem?

Answer 26

yes

Answer 27

ok one sec

Answer 28

On this page

http://www.mathplanet.com/education/geometry/transformations/transformation-using-matrices

it says that the reflection matrix for reflecting over the axis is this