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If circles are drawn taking two …

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If circles are drawn taking two sides of a triangle as diameter prove that the point of intersection of these circles lie on the third side

    • Posted by Nibha Kumari 6 years, 3 months ago

    CBSE > Class 10 > Mathematics
    • 1 answers

    Sia ? 6 years, 3 months ago

    Given: Circles are described with sides AB and AC of a triangle ABC as diameters. They intersect in a point D.
    To prove: D lies on the third side BC of {tex}\triangle{/tex}ABC.

    Construction: Join AD.
    Proof: Circle described on AB as diameter intersects BC in D.
    {tex}\angle {/tex}ADB = 90° [Angle in a semi-circle]
    But {tex}\angle {/tex}ADB + {tex}\angle {/tex}ADC = 180° [Linear Pair Axiom]
    {tex}\therefore{/tex} {tex}\angle {/tex}ADC = 90°
    Hence, the circle described on AC as diameter must pass through D.
    Thus, the two circles intersect in D.
    Now {tex}\angle {/tex}ADB + {tex}\angle {/tex}ADC = 180°
    {tex}\therefore{/tex} Points B, D, C are collinear
    {tex}\therefore{/tex} D lies on BC.

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