Consider a triangle with vertices

MATHEMATICS · Multi-part · CBSE · Mathematics · Determinants · Determinants

Problem

(x_1,y_1)

(x_2,y_2)

and

(x_3,y_3)

. (i) State the relationship between the numerical value of the determinant formed by these three points and the area of the triangle. (ii) Keeping

(x_1,y_1)

and

(x_2,y_2)

fixed, the

x

-coordinate of the third vertex is multiplied by a non-zero constant

k

while the

y

-coordinate remains unchanged. Explain how the area of the triangle changes in terms of

k

. (iii) The third vertex now moves along the line segment joining

(x_3,y_3)

to another point

(x_4,y_4)

such that its position vector is

t(x_3,y_3)+(1-t)(x_4,y_4)

, where

0\le t\le1

. Describe how the area of the resulting triangle varies with

t