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Question

The vertices of the triangle are A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D.

Find the coordinates of D and the length AD.


Solution

We know that angle bisector divides opposite side in ratio of other two sides

D divides BC in ratio of AB : AC A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2)

AB=42+52+32

=16+25+9=50+52

AC = 12+12+42

=1+1+16=18=32

AD is the internal bisector of BAC

BDDC=ABAC=53

Thus, D divides BC internally in the ratio.

D=(5×4+3×15+3,5×3+3(1)5+3,5×2+3×35+3)

D=(238,128,198)

AD=(5238)2+(4128)2+(6198)2

=172+202+29282

=289+400+84182=15308


Mathematics
RD Sharma
Standard XI

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