Question

Let A(4, 7, 8), B(2, 3, 4) and C(2, 5, 7) be the vertices of a triangle ABC. The length of the internal bisector of angle A is ______ 

Options

• `3/2sqrt34`

• `2/3sqrt34`

• `1/7sqrt34`

• `1/3sqrt34`

Answer 1

Let A(4, 7, 8), B(2, 3, 4) and C(2, 5, 7) be the vertices of a triangle ABC. The length of the internal bisector of angle A is `underline(2/3sqrt34)`.

Explanation:

In ABC, AM is the bisector of ∠BAC,

`|overline(AB)| = sqrt(4 + 16 + 16) = 6`

`|overline(AC)| = sqrt(4 + 4 + 1) = 3`

M divides BC in 2 : 1

`overlinem = (2overlinec + overlineb)/3 = 1/3(6hati + 13hatj + 18hatk)`

`overline(AM) = overlinem - overlinea = ((-6hati - 8hatj - 6hatk))/3`

l(AM) = `1/3sqrt136 = 2/3sqrt34`