The vertices of ∆ ABC are (-2, 1), (5, 4) and (2, -3) respectively. Find the area of the triangle and the length of the altitude through A.

Let three vertices be A (−2, 1) and B (5, 4) and C(2, −3)

Area of the triangle having vertices (x₁,y₁), (x₂,y₂) and (x₃,y₃)

= |x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂)|

Area of ∆ABC

= |-2(4 – (-3)) + 5(-3 -1) + 2(1 – 4)|

= |-14 - 20 - 6|

= 20 sq. units

Now to find length of BC,

By distance formula,

XY =

For BC,

BC =

=

= sq. units

Area of ∆ABC = × Base × Altitude

∴ 20 = × × Altitude

∴ Altitude = units

Hence, the length of altitude through A is units.