Two friends, Christopher and Mira, are studying the geometry of rhombuses. Christopher starts by considering the points T(5, 7) and S(7, 5) in the plane, and considers the quadrilateral OTRS, where OR = OT + OS. R 12 10 8- T 6 S 4 2 6 8 10 12 OTRS (a) Christopher determines that OT and OS are [Select all that apply]: Multiple selection advice: In a multiple selection question, marks are deducted for incorrect selections (but you cannot get less than zero for it). You are advised to only select options that you are sure about. O parallel O non-parallel O orthogonal O orthonormal O of equal magnitude O of differing magnitude Hence, Christopher deduces that OTRS is a rhombus. However, Mira then finds the diagonals of this rhombus are given by the vectors TS = and OR = and observes that TS . OR = . Hence, Mira concludes that the diagonals of OTRS Click for List Syntax advice: Enter your vectors above using Maple syntax. For example, the vector would be written as (b) However, Christopher is unsatisfied with the above result, since they have only shown that this holds for the specific rhombus in R considered above, and Mira is pretty sure they remember hearing that all rhombuses satisfy this diagonal property. Help Christopher complete the following proof: Proof. Suppose n E Z with n 2 2. Suppose A, B, C and D are points in IR" such that DC BA is a rhombus. Then the vectors Click for List represent adjacent sides of DCBA, and so are Click for List and Click for List However, by taking the dot product of the diagonals DB and CA, we get DB . CA = (DC + CB) . (CD + DA), but Click for List and Click for List , SO DB . CA = DC . DA - DC . DC + DA . DA - DA . DC. (1) (c) Christopher and Mira are now stuck and need your help finishing off this proof. Use equation (1) and the fact that DCBA is a rhombus to show that DB . CA = 0. Essay box advice: In your explanation, you don't need to use exact Maple syntax or use the equation editor, as long as your expressions are sufficiently clear for the reader. For example, you can write . A as 'A', . DCBA as 'DCBA', . DB . CA as 'DB.CA' = Q > Equation A - A - IX BIUSX X Styles Font Size Words: 0