Question

In the figure, the mid-points of the sides of the large quadrilateral are joined to draw the small quadrilateral inside. Then the coordinates of points $P$ and $Q$ respectively are

• A. $(4,1)$ and $(6,4)$
• B. $(6,4)$ and $(4,1)$
• C. $(4,2)$ and $(2,4)$
• D. $(2,4)$ and $(4,2)$

Answer 1

The correct option is B

$(6,4)$ and $(4,1)$

Let the vertices of the given quadrilateral be $A(0,0),B(x_1,y_1),P(x_2,y_2)$ and $D(x_3,y_3)$.

Also, let the unknown vertex of the smaller quadrilateral formed by joining mid-points of sides of the given quadrilateral be $Q(a,b)$.

We know that the mid-point of the line joining two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$.

Now, applying mid-point formula for the line segment AD, we get,

$(2,2)=(\frac{0+x_3}{2},\frac{0+y_3}{2})$

$⟹(2,2)=(\frac{x_3}{2},\frac{y_3}{2})$

$⟹x_3=4$ and $y_3=4$

Therefore coordinates of point D are $(4,4)$

Similarly, applying the mid-point formula for line segment DP, we get,

$(5,4)=(\frac{4+x_2}{2},\frac{4+y_2}{2})$

$⟹x_2=6$ and $y_2=4$

Therefore coordinates of point P are $(6,4)$

Again, by applying mid-point formula for the line segment PB, we get,

$(7,3)=(\frac{6+x_1}{2},\frac{4+y_1}{2})$

$⟹x_1=8$ and $y_1=2$

Therefore coordinates of point B are $(8,2)$

Now, we shall apply mid-point formula for the line segment AB so as to find the coordinates of point Q.

$(a,b)=\frac{0+8}{2},\frac{0+2}{2})$

$⟹a=4$ and $b=1$

Therefore coordinates of point Q are $(4,1)$.