In the diagram, ABDE is a rectangle, ABCD is equilateral, and AD is parallel to BC. Also, AE = 2x for some real number x. (i) Determine the length of AB in terms of x. (ii) Determine positive integers r and s for which AC/AD =square root r/s (a) Suppose that a = 5 and b = 4. Determine all pairs of integers (K, L) for which K2 + 3L2= a2 + b2 − ab. (b) Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 + 3L2 = a2 + b2 − ab. (c) Prove that, for all integers a and b, there is at least one pair of integers (K, L) for which K2 + 3L2 = a2 + b2 − ab.

Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
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  1. In the diagram, ABDE is a rectangle, ABCD
    is equilateral, and AD is parallel to BC. Also,
    AE = 2x for some real number x.
    (i) Determine the length of AB in terms of x.
    (ii) Determine positive integers r and s for which
    AC/AD =square root r/s

  2. (a) Suppose that a = 5 and b = 4. Determine all pairs of integers (K, L) for which K2 + 3L2= a2 + b2 − ab.

    (b) Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 + 3L2 = a2 + b2 − ab.

    (c) Prove that, for all integers a and b, there is at least one pair of integers (K, L) for which K2 + 3L2 = a2 + b2 − ab.

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