Question

(a) Plot the points $P(0,3),Q(3,0),\mathrm{and}R(6,3)$in the co-ordinate plane. Where must the point S be located so that PQRS is a square?

(b) Find the area of PQRS.

Short answer

(a) Co-ordinate of S is (3,6).

(b) Area of a square$=18$

Step-by-step solution

Part a. Step 1. Property of a square.

The diagonal of a square bisects each other.

Part a. Step 2. Given condition.

Vertex of the square PQRS is $P(0,3),Q(3,0),R(6,3)\mathrm{and}S(h,k)$.

Diagonals of the square are PR and QS.

Co-ordinate of mid-point of diagonal is$\left(\frac{6+0}{2},\frac{3+3}{2}\right)=(3,3)$

Part a. Step 3. Finding the point.

Mid-point of Q and S is $\left(\frac{3+h}{2},\frac{0+k}{2}\right)=(3,3)$

$$\begin{gathered} 3+h=6,k=6 \\ h=6-3 \\ h=3 \end{gathered}$$

So, Co-ordinate of S is (3,6).

Part b. Step 1. Formula of area of square.

Area of square$={\left(\mathrm{side}\right)}^2$

Part b. Step 2. Determining the side of a square.

Here,$P=(0,3)\mathrm{and}Q=(3,0)$

$$\begin{gathered} \mathrm{So},PQ=\sqrt{3^2+3^2} \\ =3\sqrt{2} \end{gathered}$$

So, Side of square$=3\sqrt{2}$

Part b. Step 3. Determining the area of square.

$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{square}={\left(3\sqrt{2}\right)}^2 \\ =18 \end{gathered}$$