Find the area of the Triangle ABC, with vertex A(1,-4) and midpoint of AB and AC as X(2,-1) and Y(0,-4) respectively.

 

Asked by nijurajeev7276 | 8th Jan, 2020, 09:52: AM

Expert Answer:

To find area of the triangle ABC, with vertex A(1,-4) and midpoint of AB and AC as X(2,-1) and Y(0,-4) respectively.
Let B(x,y) and C(a, b)
Midpoint of AB is (2, -1)
rightwards double arrow open parentheses fraction numerator straight x plus 1 over denominator 2 end fraction comma fraction numerator straight y minus 4 over denominator 2 end fraction close parentheses equals open parentheses 2 comma space minus 1 close parentheses
rightwards double arrow straight x plus 1 equals 4 space and space straight y minus 4 equals negative 2
rightwards double arrow straight x equals 3 space and space straight y equals 2
Also comma space midpoint space of space AC space is space left parenthesis 0 comma negative 4 right parenthesis
rightwards double arrow open parentheses fraction numerator straight a plus 1 over denominator 2 end fraction comma fraction numerator straight b minus 4 over denominator 2 end fraction close parentheses equals open parentheses 0 comma space minus 4 close parentheses
rightwards double arrow straight a plus 1 equals 0 space and space straight b minus 4 equals negative 8
rightwards double arrow straight a equals negative 1 space and space straight b equals negative 4
Thus comma space the space coordinates space of space straight B space and space straight C space are space left parenthesis 3 comma space 2 right parenthesis space and space left parenthesis negative 1 comma space minus 4 right parenthesis space respectively.
Area space of space triangle ABC equals 1 half open vertical bar open square brackets straight x subscript 1 open parentheses straight y subscript 2 minus straight y subscript 3 close parentheses plus straight x subscript 2 open parentheses straight y subscript 3 minus straight y subscript 1 close parentheses plus straight x subscript 3 open parentheses straight y subscript 1 minus straight y subscript 2 close parentheses close square brackets close vertical bar
equals 1 half open vertical bar open square brackets 1 open parentheses 2 plus 4 close parentheses plus 3 open parentheses negative 4 plus 4 close parentheses minus 1 open parentheses negative 4 minus 2 close parentheses close square brackets close vertical bar
equals 1 half open vertical bar open square brackets 6 plus 0 plus 6 close square brackets close vertical bar equals 6
Hence comma space area space of space the space triangle space ABC thin space is space 6 space sq space units.

Answered by Renu Varma | 8th Jan, 2020, 10:40: AM