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State true or false.

In $Δ A B C$ , $∠ B = 35 °, ∠ C = 65 °$ and the bisector of $∠ B A C$ meets $B C$ in $X$ . Then the descending order of sides is $B X, C X, A X$ .

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True

False

answer is B.

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Detailed Solution

It is given that

∠ B = 35 °, ∠ C = 65 °

.
Consider

Δ A B C

given below.
Question Image

By angle sum property of a triangle,

∠ A + ∠ B + ∠ C = 180 °

∠ A + 35 ° + 65 ° = 180 °

∠ A = 180 ° − 100 °

= 80 °

But,

∠ B A X = 12 ∠ A

∠ B A X = 12 × 80 °

= 40 °

Consider

Δ A B X

∠ B = 35 °

and

∠ B A X = 40 °

.
By angle sum property of the triangle,

∠ B X A + ∠ B + ∠ B A X = 180 °

∠ B X A + 35 ° + 40 ° = 180 °

∠ B X A = 180 ° − 75 °

= 105 °

Δ A B X

∠ B

is the smallest angle.
The side

A X

opposite to it will be the smallest.

∴ A X < B X

Consider

Δ A C X

∠ C A X = 12 ∠ A

∠ C A X = 12 × 80 °

= 40 °

And, in

∠ A X C + ∠ C A X + ∠ C = 180 °

∠ A X C + 40 ° + 65 ° = 180 °

∠ A X C = 180 ° − 105 °

= 75 °

Δ A C X, ∠ C A X = 40 °,

∠ C = 65 °

∠ A X C = 75 °

So,

∠ C A X

is the smallest angle.
The side

C X

opposite to it is the shortest side.
Therefore,

C X < A X

.
Therefore, the given sides in descending order

B X > A X > C X

.
Hence the statement is not true.
The correct statement is the descending order of sides is BX, AX, CX.
Therefore option 2 is correct.

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