The bisector of angle A of triangle ABC meets BC at D such that AD is perpendicular to BC. If AB = (2x + y) cm, BD = (3x + 1) cm, BC = (3x + 2y ─ 1) cm and AC = (8x ─ y) cm, then the values of x and y are:
Answers
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Given :-
- The bisector of angle A of triangle ABC meets BC at D such that AD is perpendicular to BC.
- AB = (2x + y) cm.
- BD = (3x + 1) cm.
- BC = (3x + 2y ─ 1) cm
- AC = (8x ─ y) cm.
To Find :-
- find x and y ?
Concept used :-
- if angle bisector of the vertex angle is also the perpendicular bisector of the base of a triangle, than, given triangle is an Isosceles Triangle .
Solution :-
Given that, AD is angle bisector and perpendicular to BC. .
Therefore, ∆ABC is an Isosceles Triangle.
Hence,
→ AB = AC
→ (2x + y) = (8x - y)
→ y + y = 8x - 2x
→ 2y = 6x
→ y = 3x
Now,
→ DC = BC - BD
→ DC = (3x + 2y - 1) - (3x + 1)
Putting value of y = 3x
→ DC = (3x + 2*3x - 1) - 3x - 1
→ DC = 9x - 1 - 3x - 1
→ DC = (6x - 2)
Now, since,
→ BD = DC { AB = AC .}
→ 3x + 1 = 6x - 2
→ 1 + 2 = 6x - 3x
→ 3x = 3
→ x = 1. (Ans.)
Hence,
→ y = 3 . (Ans.)
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