A commuter airline company, Geometry Central, currently serves four cities; Addi

Question

A commuter airline company, Geometry Central, currently serves four cities; Addison, Braxton, Clayton, and Davidson – labeled A, B, C, and D respectively. Geometry Central has hired you to build their new hub in a more centrally located area where service between all four cities will be reduced. You have determined the location of the new hub to be point H. In addition, the hub will add 4 new routes to the schedule:   AH, BH, CH, and DH .

1. How could you apply the Triangle Inequality Theorem to write an inequality involving AX ,CX and AC; assuming X is not the same point as H? How could you rewrite the inequality to include AH and CH instead of AC?

2. The executives at Geometry Central want to know why your location at point H (AH + BH + CH + DH) is the smallest combined distance between the cities. Give an indirect proof or argument in favor to explain why the airline should build the new hub at the intersection of the diagonals AC and BD . (Hint: Assume a different point X results in a smaller combined distance. Then show how AX + CX compares to AH + CH.)

3. How can you find AC? What is the relationship between AB and BC? Round your answer to the nearest tenth.

4. How are DB and DC related? What are their measures? Round your answer to the nearest tenth.

5. Presently, travelers who want to fly from Addison to Clayton must fly through Braxton. After the new hub is built, travelers will have a shorter route from Addison to the new hub and then to Clayton. How many miles will this save? Round your answer to the nearest tenth.

6. Presently, travelers who want to fly from Braxton to Davidson must fly through Clayton. After the new hub is built, travelers will have a shorter route from Braxton to the new hub and then to Davidson. How many miles will this save? Round your answer to the nearest tenth.

7. Only comparing the sides of quadrilateral ABCD, which route appears to be the longest? How could you show that this route is longer than AD ?

8. After the new airport hub is built, Geometry Central will serve a meal on its longest non-stop flight. On which route should they serve the meal? How do you know this is the longest route? Write a paragraph proof.

9. Use triangle ACD to find the range of possible distances for the route from Addison to Davidson.

Explanation / Answer

In a triangle ACX where A represents Addison and C rpresents Clayton and X is not the hub H

THe triangle inequality theorem states that the sun of any two sides of the triangle ACX is greater then the third side. THe inequalities are :

AX + CX > AC

AX + AC > CX

CX + AC > AX

now as H lies on the line AC so H divides AC in two parts and they are AH and CH.

so the inequalities could be writen as :

AX + CX > AH + CH

AX + AH + CH > CX

CX + AH + CH > AX

3>

the triangle ABC is rightangled at B and the angle A is = 45 degrees

AB = 200 miles

=> Cos(45 degrees) = AB/AC = 200/AC

=> AC = 200/Cos(45 degrees) = 200sqrt(2) = 282.8 miles

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