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  • In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see Fig.). Show that: (ii) ∠ DBC is a right angle.

8. In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that \small DM=CM. Point D is joined to point B (see Fig.). Show that: \small \angle DBC is a right angle. 

    

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In the previous part, we have proved that   \small \Delta AMC\cong \Delta BMD.

By c.p.c.t. we can say that :  \angle ACM\ =\ \angle BDM

This implies side AC is parallel to BD.

Thus we can write :  \angle ACB\ +\ \angle DBC\ =\ 180^{\circ}    (Co-interior angles)

and, 90^{\circ}\ +\ \angle DBC\ =\ 180^{\circ}

or \angle DBC\ =\ 90^{\circ}

Hence \small \angle DBC is a right angle.

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Sanket Gandhi

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