In figur bisectors of angle B and angle C of ∆ABC intersects each other in point X. Line AX intersects side BC in point Y . AB=5,AC=4, BC=6 then find AX/XY.
Answers
Answer:
AX / XY = 3 / 2
Step-by-step-explanation:
NOTE: Refer to the attachment for the diagram.
In figure, In △ABY,
Ray BX bisects ∠ABY. - - [ Given ]
∴ By angle bisector theorem of triangle,
AB / BY = AX / XY - - ( 1 )
Now,
In △ACY,
Ray CX bisects ∠ABY. - - [ Given ]
∴ By angle bisector theorem of triangle,
AC / CY = AX / XY - - ( 2 )
From ( 1 ) & ( 2 )
AB / BY = AC / CY = AX / XY
⇒ 5 / BY = 4 / CY = AX / XY
Now, by applying the theorem on equal ratios, we get,
( 5 + 4 ) / ( BY + CY ) = AX / XY
⇒ 9 / BC = AX / XY - - - [ B - Y - C ]
⇒ 9 / 6 = AX / XY
⇒ AX / XY = 3 / 2
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Additional Information:
1. Angle bisector theorem:
When a ray bisects an angle, every point on the ray is equidistant from the both arms of the angle.
2. Angle bisector theorem of triangle:
When a ray bisects an angle of a triangle, the arms of the angle and the remaining two sides are in the proportion.
3. This theorem is based on Basic Proportionality Theorem ( BPT ).
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Answer
In fig, in ∆ AB¥,
- Ray BX bisects ∆AB¥, --->[Given]
- By angle bisector theorem of triangle,
AB / B¥ = AX X¥ ----> (1)
- Now,
In ∆AC¥
- Ray CX bisects ∆AB¥, ----->[Given]
- By angle bisector theorem of triangle,
AC / C¥ = AX / X¥ ----> (2)
- from eqn (1) and (2)
AB / B¥ =AC / C¥ =AX / XY
➜5 / B¥ = 4 / C¥ = AX / X¥
- Now applying the theorem on equal ratios,we get,
➜(5 / 4) / (B¥ / C¥) = AX / X¥
➜ 9 / BC = AX / X¥ --(B-X-C)
➜ 9 / 6 = AX / X¥
➜ AX / X¥ = 2 / 3
