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Thread: Isosceles Triangle ABC

  1. #1
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    Isosceles Triangle ABC

    Given: Isosceles triangle ABC with segment AB congruent to segment AC, and line CD is perpendicular to AB.

    Prove: Segment BC multiplied by segment BC is equal to two times Segment AB multiplied by Segment BD.

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  2. #2
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    Hello sologuitar
    Quote Originally Posted by sologuitar View Post
    Given: Isosceles triangle ABC with segment AB congruent to segment AC, and line CD is perpendicular to AB.

    Prove: Segment BC multiplied by segment BC is equal to two times Segment AB multiplied by Segment BD.

    Using Pythagoras' Theorem on $\displaystyle \triangle ADC$:
    $\displaystyle DC^2=AC^2-AD^2$
    $\displaystyle = AB^2 -AD^2$, since $\displaystyle AC = AB$ (given)

    $\displaystyle =(AB+AD)(AB-AD)$

    $\displaystyle =(AB+AD)(BD)$ ... (1)
    And using Pythagoras' Theorem on $\displaystyle \triangle BDC$:
    $\displaystyle BC^2 = BD^2+DC^2$
    $\displaystyle = BD^2 + (AB+AD)(BD)$, from (1)

    $\displaystyle =BD(BD +AB+AD)$

    $\displaystyle =BD(AB+AB)$

    $\displaystyle =2BD.AB$
    Grandad
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