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ABC is a right angled triangle angle ABC is a right angle, a bisector of angle ABC intersect AC at D. measuring AB=3 cm, BC=4 cm, find AD ?



From internal angle bisector theorem, AD:DC=3:4, since ∆ABC is right angled at B, AB²+BC²=AC². Hence, 3²+4²=(7a)², a=5/7. Therefore, AD=15/7 cm.
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In right angled triangle ABC, (AB)2 + (BC)2 = (AC)2 Therefore, AC = 5 cm. ...By Pythagoras Theorem In triangle ABC, AD is the angle bisector Therefore, AB/BC = AD/DC. ....Property of an angle bisector of a triangle Therefore, BC/AB = DC/AD. ...By Invertendo Therefore, BC+AB/AB = DC+AD/AD ...By Compenendo Therefore, 4+3/3 = AC/AD . 7/3 = 5/AD . AD= 5*3/7 . AD= 2.14cm
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