Answer: The co-ordinates of F are (4.4, -4.6) and the co-ordinates of D are (4.4, -1.4).
Step-by-step explanation: Given that a square is constructed on side AD of quadrilateral ABCD such that FA lies on AB as shown in the figure. The co-ordinates of A are (6, -3) and the co-ordinates of B are (10, 1).
Also, AD : AB = 2 : 5 and the co-ordinates of the point E are (2.8, -3).
We are to select the correct co-ordinates of the points F and D.
Let, (a, b) are the co-ordinates of F and (c, d) are the co-ordinates of D.
Since ADEF is a square, so we have
AD = DE = EF = FA.
Given that
AD : AB = 2 : 5, so FA : AB = 2 : 5.
That is, \left(\dfrac{c+4.4}{2},\dfrac{d-4.6}{2}\right)=\left(\dfrac{2.8+6}{2},\dfrac{-3-3}{2}\right)
We have, after applying the internal division formula that
[tex]\left(\dfrac{2\times 10+5\times a}{2+5},\dfrac{2\times 1+5\times b}{2+5}\right)=(6,-3)\\\\\\\Rightarrow \left(\dfrac{20+5a}{7},\dfrac{2+5b}{7}\right)=(6,-3)\\\\\\\Rightarrow \dfrac{20+5a}{7}=6,~~~~~\dfrac{2+5b}{7}=-3\\\\\\\Rightarrow 20+5a=42,~~~~\Rightarrow 2+5b=-21\\\\\\\Rightarrow 5a=22,~~~~~~~~~~\Rightarrow 5b=-23\\\\\\\Rightarrow a=4.4,~~~~~~~~~~~\Rightarrow b=-4.6.[/tex]
So, the co-ordinates of F are (4.4, -4.6).
Now, since ADEF is a square, and the diagonals of a square bisect each other.
So, the mid-points of both the diagonals are same.
That is,
[tex]\textup{mid-point of DF}=\textup{mid-point of AE}\\\\\\\Rightarrow \left(\dfrac{c+4.4}{2},\dfrac{d-4.6}{2}\right)=\left(\dfrac{2.8+6}{2},\dfrac{-3-3}{2}\right)\\\\\\\Rightarrow \left(\dfrac{c+4.4}{2},\dfrac{d-4.6}{2}\right)=\left(\dfrac{8.8}{2},\dfrac{-6}{2}\right)\\\\\\\Rightarrow \dfrac{c+4.4}{2}=\dfrac{8.8}{2},~~~~~~\dfrac{d-4.6}{2}=-\dfrac{6}{2}\\\\\\\Rightarrow c+4.4=8.8,~~~~~\Rightarrow d-4.6=-6\\\\\Rightarrow c=4.4,~~~~~~~~~~~~\Rightarrow d=-1.4.[/tex]
So, the co-ordinates of D are (4.4, -1.4).
Thus, the co-ordinates of F are (4.4, -4.6) and the co-ordinates of D are (4.4, -1.4).
Answer:
F is (4.4, -4.6) and D is (4.4, -1.4).
Hope this helps :)
Step-by-step explanation:
