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

2. 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.

3. 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.

4. 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 ?

5. 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.

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

Explanation / Answer

1. As DCB is a right angled triangle and DB is the hypotenuse, DB>DC. As ABC is the right angle triangle and angle A=45, we have tan 45 =BC/AB and so BC=200mi. Similarly from triangle BCD, we have tan 30 = BC/DC and so CD = 350 (nearest tenth). As DB is the hypotenuse of BCD, we have DB=400mi.

2.We have AC=280mi (as AC is the hypotenuse of triangle ABC, AB=200mi and BC=200mi). Therefore the route Addison to new hub and then to Clayton saves AB+BC-AC=120mi.

3. We have BD=400mi, BC=200mi and CD=350mi. Therefore the route Braxton to Davidson via new hub saves BC+CD-BD=150mi

4.As the route AD=250mi, the route CD is the longest among the sides of the Quadrilateral. Draw a line perpendicular to CD from A to E(say on line CD), which has length 200mi as it is parallel to BC. AD is the hypotenuse of triangle ADE, AD has the longest length among the sides AD, AE and DE. Now clearly as CE = AE, we have CD=AE+ED > AD and it is clear that AD is the hypotenuse of triangle AED with AE=200mi, ED=150mi, we have AD=250mi.

5. After a new hub is build, the Geometry Central will serve a meal on the route BD, which is of the largest distance equal to 400mi. As in the previous subdivision draw a line perpendicular to CD from A to E(say on CD) of length 200mi. As ABCE is a square and AC is a diagonal to it, we have DB>AC. The only thing to compare with the side DB is the length of the side CD. Let the point of intersection of the line AE with the line DB be F. Then as DEF is a right angle triangle, we have DF>DE and so DF+CE>DE+EC=DC. But as FB>EC(as it is more like a diagonal of a square ABCE), we have DF+FB>DC.

6. The range is 250mi to 630mi, as AD is 250mi and AC+CD=280+350=630mi.

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