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Question

In figure, ABC is a triangle with B=35,C=65 and bisector of BAC meets BC in X. Arrange AX,BX and CX in descending order.
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Solution

Consider ABC
By sum property of a triangle
A+B+C=180
To find A
A=180BC
By substituting the values
A=1803565
By subtraction
A=180100
A=80
We know that
BAX=12A
So we get
BAX=12(80)
By division
BAX=40
Consider ABX
It is given that B=35 and BAX=40
By sum property of a triangle
BAX+BXA+XBA=180
To find BXA
BXA=180BAXXBA
By substituting values
BXA=1803540
By subtraction
BXA=18075
BXA=105
We know that B is the smallest angle and the side opposite to it i.e. AX is the smallest side.

So we get AX<BX..(1)
Consider AXC
CAX=12A
So we get
CAX=12(80)
By division
CAX=40
By sum property of a triangle
AXC+CAX+CXA=180
To find AXC
AXC=180CAXCXA
By substituting values
AXC=1804065
So we get
AXC=180105
By subtraction
AXC=75
So we know that CAX is the smallest angle and the side opposite to it i.e. CX is the smallest side.
We get
CX<AX(2)
By considering equation (1) and (2)
BX>AX>CX
Therefore, BX>AX>CX is the descending order.

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