Class 10

Math

All topics

Coordinate Geometry

Find the area of $ΔABC$ with $A(1,−4)$ and midpoints of sides through A being $(2,−1)$ and $(0,−1)$.

Given: A $ΔABC$ with $A(1,−4)$

Let F and E are the midpoints of AB and AC respectively

Let Coordinates of F are $(2,−1)$ and Coordinates of E are $(0,−1)$

Let Coordinates of B be $(x_{1},y_{1})$ and Coordinates of C be $(x_{2},y_{2})$

Find coordinate of B:

using section formula:

$2=21+x_{1} ⇒x_{1}=4−1=3$

$−1=24+y_{1} ⇒y_{1}=−2+4=2$

Coordinate of B are $(3,2)$

Find coordinate of C:

using section formula:

$0=21+x_{2} ⇒1+x_{2}=0⇒x_{2}=−1$

$−1=2−4+y_{2} ⇒−4+y_{2}=−2$

$⇒y_{2}=−2+4=2$

Coordinate of C are $(−1,2)$

Now,

Area of $ΔABC=21 [x_{1}(y_{2}−y_{3})+x_{2}(y_{3}−y_{1})+x_{3}(y_{1}−y_{2})]$

Area of triangle ABC:

$=21 [1(2−2)+3(2+4)+(−1)(−4−2)]$

$=21 [1×0+3×6+(−1)×(−6)]$

$=21 [0+18+6]=224 =12$ sq. units.