If (-2.3), (4.-3) and (4,5) are the mid points of the sides of a triangle, then find the coordinates of triangle.
Answers
Step-by-step explanation:
Given :-
(-2,3), (4,-3) and (4,5) are the mid points of the sides of a triangle.
To find :-
Find the coordinates of triangle?
Solution:-
Given that
Mid points of the sides of a triangle are (-2,3), (4,-3) and (4,5)
Let the sides of a triangle are A,B,C
Let A(x1, y1)
Let B(x2, y2)
Let C(x3, y3)
Let the mid point of AB = D(-2,3)
Let the mid point of BC =E(4,-3)
Let the mid point of AC = F(4,5)
We know that
The coordinates of the mid-point of the line segment joining the points (x1, y1) and (x2, y2) is
( (x1+x2)/2,(y1+y2)/2 )
I) Mid point of AB =( (x1+x2)/2,(y1+y2)/2 ) = (-2,3)
=> (x1+x2)/2 = -2 and (y1+y2)/2 = 3
=> x1+x2 = 2×-2 and y1+y2 = 2×3
=> x1+x2 = -4 ---------(1)
and y1+y2 = 6 --------(2)
II) Mid point of BC = ( (x2+x3)/2,(y2+y3)/2 ) = (4,-3)
=> (x2+x3/2 = 4 and (y2+y3)/2 = -3
=> x2+x3 = 4×2 and y2+y3 = 2×-3
=> x2+x3 = 8 -----------(3)
and y2+y3 = -6 --------(4)
III) Mid point of AC =( (x1+x3)/2 , (y1+y3)/2) = (4,5)
=> (x1+x3)/2 = 4 and (y1+y3)/2 = 5
=> x1+x3 = 2×4 and y1+y3 = 2×5
=> x1+x3 = 8 --------------(5)
and y1+y3 = 10 -----------(6)
On adding (1),(3),(5) equations then
=>x1+x2+x2+x3+x1+x3 = -4+8+8
=> 2x1+2x2+2x3 = 16-4
=> 2(x1+x2+x3) = 12
=> x1+x2+x3 = 12/2
=>x1+x2+x3 = 6
From (3)
=> x1+8 = 6
=>x1 = 6-8
=> x1 = -2
and from (1)
x1+x2 = -4
=>-2+x2 = -4
=> x2 = -4+2
=> x2 = -2
from (3)
x2+x3 = 8
=> -2+x3 = 8
=> x3 = 8+2
=>x3 = 10
and
On adding (2),(4),(6)
=> y1+y2+y2+y3+y3+y1 = 6-6+10
=> 2y1+2y2+2y3 = 10
=> 2(y1+y2+y3) = 10
=> y1+y2+y3 = 10/2
=> y1+y2+y3 = 5
from (2)
=> 6+y3 = 5
=> y3 = 5-6
=> y3 = -1
From (6)
y2+10 = 5
=> y2 = 5-10
=> y2 = -5
From (6)
y1-1 = 10
=> y1 = 10+1
=> y1 = 11
We have, x1 = -2 , x2 = -2 ,x3 = 10
and y1 = 11 ,y2 = -5 , y3 = -1
A =(-2,11) , B(-2,-5) , C(10,-1)
Answer :-
The coordinates of the three sides of the triangle are:
The coordinates of A =(-2,11)
The coordinates of B = (-2,-5)
The coordinates of C = (10,-1)
Used formulae:-
The coordinates of the mid-point of the line segment joining the points (x1, y1) and (x2, y2) is ( (x1+x2)/2,(y1+y2)/2 )
