Multiple Choice Question (MCQ)

In the figure, given coordinates of point A is (3,4) in the rectangular coordinates. Let point P is a dynamic point on the positive x axis such that \triangle AOP is an isosceles triangle. How many such triangle could be formed?

  1. ×

    1

  2. ×

    2

  3. 3

  4. ×

    4

Collected in the board: Coordinate geometry

Steven Zheng posted 1 week ago

Answer

  1. Moving the point P along the x axis finds 3 isosceles triangles could be formed.

    Since point A is a fixed point, using the distance formula between two points gives

    AO = \sqrt{3^3+4^2}

    \cos \angle AOP = \dfrac{3}{5}

    1. When AP = OP

    OP = \dfrac{OD}{\cos \angle AOP}


    = \dfrac{\dfrac{AO}{2} }{\cos \angle AOP}


    = \dfrac{2.5}{\dfrac{3}{5} } = \dfrac{5}{2}\cdot \dfrac{5}{3}=\dfrac{25}{6}

    The coodinates of P is (\dfrac{25}{6} ,0)

    2. When OP = OA = 5

    The coodinates of P is (5,0)

    3. When OP = AP

    The coodinates of P is (6,0)

Steven Zheng posted 1 week ago

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