Anonymous
Anonymous asked in Science & MathematicsMathematics · 3 years ago

丁丑  Minimum of AP*BP = ___.?

Line segment [AB = 7] is on a plane.

Distance of point P and AB is 3.

Minimum of AP*BP = ___.

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  • atsuo
    Lv 6
    3 years ago
    Favorite Answer

    Think △APB , its base is AB .

    The height of △APB is the distance between P and AB , so it is 3 .

    Therefore , the area of △APB is S = (1/2)7*3 = 21/2 .

    On the other hand , the area of △ABP must be S = (1/2)AP*BP*sin(∠APB) .

    So

    (1/2)AP*BP*sin(∠APB) = 21/2

    AP*BP*sin(∠APB) = 21

    AP*BP = 21 / sin(∠APB)

    And 0 < sin(∠APB) ≦ 1 , therefore

    AP*BP has the minimum value of 21 when ∠APB = 90° .

    (The lengths of AP and BP are not requested .)

    • JOHN
      Lv 7
      3 years agoReport

      Nice!

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  • 3 years ago

    Minimum of AP*BP = (√((7/2)² + 3²) )² = 49/4 + 9 = 85/4

    • ...Show all comments
    • JOHN
      Lv 7
      3 years agoReport

      Naive intuition sees only one symmety, which is what caused this answer to go wrong. Symmetrry arguments can be formulated rigorously, and are mathematically valid.

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