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# Geometry

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A music company has a stores in Abby (-3, -2) and Cardenas (3, 6). Find each unit of the coordninate plane represents 1 mile.

a. The company president wants to build a warehouse that is equdistant from the two stores. Write an equation that describes the possible locations.

b. A straight road connects Abby and Cardenas.. The warehouse will be located exactly 4 miles from the road. How many locations are possible?

c. To the nearest tenth of a mile, how far will the warehouse be from each store?

Jan 24, 2018
edited by NiteWolf813  Jan 24, 2018

#1
+7564
+1

a.

distance between warehouse and Abby   =   distance between warehouse and Cardenas

Let  (x, y)  be the coordinates of the warehouse.

distance between  (x, y)  and  (-3, -2)   =   distance between  (x, y)  and  (3, 6)

√[  (x + 3)2 + (y + 2)2 ]   =   √[  (x - 3)2 + (y - 6)2  ]

(x + 3)2 + (y + 2)2   =   (x - 3)2 + (y - 6)2

x2 + 6x + 9  +  y2 + 4y + 4   =   x2 - 6x + 9  +  y2 - 12y + 36

6x + 4y + 4   =   -6x - 12y + 36

16y   =   -12x + 32

y  =  -3/4x + 2         The warehouse can be any point along this line.

Here's a graph:      https://www.desmos.com/calculator/eenwtk2dms

b.

The warehouse needs to be on the line   y = -3/4x + 2   and needs to be  4  units away from the road. So there are two possible locations.

*edit*

c.

I think the answer to this part is  6.4  miles, but I don't have a good explanation.

Jan 24, 2018
edited by hectictar  Jan 24, 2018

#1
+7564
+1

a.

distance between warehouse and Abby   =   distance between warehouse and Cardenas

Let  (x, y)  be the coordinates of the warehouse.

distance between  (x, y)  and  (-3, -2)   =   distance between  (x, y)  and  (3, 6)

√[  (x + 3)2 + (y + 2)2 ]   =   √[  (x - 3)2 + (y - 6)2  ]

(x + 3)2 + (y + 2)2   =   (x - 3)2 + (y - 6)2

x2 + 6x + 9  +  y2 + 4y + 4   =   x2 - 6x + 9  +  y2 - 12y + 36

6x + 4y + 4   =   -6x - 12y + 36

16y   =   -12x + 32

y  =  -3/4x + 2         The warehouse can be any point along this line.

Here's a graph:      https://www.desmos.com/calculator/eenwtk2dms

b.

The warehouse needs to be on the line   y = -3/4x + 2   and needs to be  4  units away from the road. So there are two possible locations.

*edit*

c.

I think the answer to this part is  6.4  miles, but I don't have a good explanation.

hectictar Jan 24, 2018
edited by hectictar  Jan 24, 2018
#2
+100483
+2

Here's an explanation for  "c"

We have two possible congruent right triangles.....

The midpoint   of these two coordinates is  (0,2) ......the distance from this point to either store is

sqrt [ (3- 0)^2 + (6 - 2)^2 ]  =  sqrt (9 + 16)  = sqrt (25)  =  5 miles

And this is one leg of either right triangle......and the other leg is the distance from the warehouse to the road  = 4 miles

So.....the distance from  the warehouse to either store is the hypotenuse of these triangles

sqrt (  5^2  + 4^2)  =  sqrt ( 41)  =  6.4 miles

JUST AS HECTICTAR FOUND....!!!!

Jan 24, 2018
edited by CPhill  Jan 24, 2018