# Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4) - Mathematics and Statistics

Sum

Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4)

#### Solution

Let bar"a", bar"b", bar"c" be the position vectors of points A, B, C respectively of ∆ABC and bar"h" be the position vector of its incentre H.

∴ bar"h" = (|bar"BC"|bar"a" + |bar"AC"|bar"b" + |bar"AB"|bar"c")/(|bar"BC"| + |bar"AC"| + |bar"AB"|)     ...(i)

∴ bar"a" = 3hat"j", bar"b" = 4hat"k", bar"c" = 3hat"j" + 4hat"k"

∴ bar"BC" = bar"c" - bar"b" = (3hat"j" + 4hat"k") - 4hat"k" = 3hat"j"

bar"AC" = bar"c" - bar"a" = (3hat"j" + 4hat"k") - 3hat"j" = 4hat"k"

bar"AB" = bar"b" - bar"a" = 4hat"k" - 3hat"j"

∴ |bar"BC"| = sqrt(9) = 3

|bar"AC"| = sqrt(16) = 4

and

|bar"AB"| = sqrt(16 + 9) = sqrt(25) = 5

∴ bar"h" = (3(3hat"j") + 4(4hat"k") + 5(3hat"j" + 4hat"k"))/(3 + 4 + 5)  .......[From (i)]

∴ bar"h" = (9hat"j" + 16hat"k" + 15hat"j" + 20hat"k")/12

= (24hat"j" + 36hat"k")/12

= 2hat"j" + 3hat"k"

∴ Incentre of the triangle is H (0, 2, 3).

Concept: Section Formula
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