question_answer 1)

Avneet buys 9 square paving slabs, each with a side of \[\frac{1}{2}m.\] He lays them in the form of a square. (a) What is the perimeter of his arrangement in the figure (i)? (b) Shari does not like his arrangement. She gets him to lay them out like a cross. What is the perimeter of her arrangement in the figure (ii)? (c) Which has greater perimeter? (d) Avneet wonders, if there is a way of getting an even greater perimeter. Can you find a way of doing this? (The paving slabs must meet along complete edges i.e. they cannot be broken.)

Answer:

(a) Avneet lays 9 squares in the form of a square as shown in the figure, then side of the square \[=\left( \frac{1}{2}+\frac{1}{2}+\frac{1}{2} \right)m=\frac{1+1+1}{2}=\frac{3}{2}m\] \[\therefore \]Perimeter of Avneet's arrangement \[=4\times \] Length of a side \[=4\times \frac{3}{2}m=6m\] Hence, the perimeter of Avneet's arrangement is 6 m. (b) Shari lays 9 squares in the form of a cross as shown in figure, then perimeter of Shari's arrangement = Sum of all sides \[=AB+BC+CD+DE+EF+FG\] \[+GH+HI+IJ+JK+KL+LA~~~~~~~\] \[=\frac{1}{2}+\left( \frac{1}{2}+\frac{1}{2} \right)+\left( \frac{1}{2}+\frac{1}{2} \right)+\frac{1}{2}+\left( \frac{1}{2}+\frac{1}{2} \right)+\left( \frac{1}{2}+\frac{1}{2} \right)\] \[=\left( \frac{1}{2}+1+1+\frac{1}{2}+1+1+\frac{1}{2}+1+1+\frac{1}{2}+1+1 \right)m\] \[=\left( \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+8 \right)m=(1+1+8)m=10m\] Hence the perimeter of Shari's arrangement is 10 m. (c) From above, it is clear that Shari's arrangement i.e. a cross has greater perimeter. (d) Yes, there is a way shown alongside in which we get a greater perimeter. Here we have arrange the 9 squares in the form of a rectangle Now, length of rectangle \[=\left( \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2} \right)m=\frac{9}{2}m\] and breadth of rectangle = 1/2 m \[\therefore \] Perimeter of this rectangle \[=2\times \] (Length + Breadth) \[=2\times \left( \frac{9}{2}\times \frac{1}{2} \right)=2\times \left( \frac{9+1}{2} \right)=2\times \frac{10}{2}=10m\] It is clear that, it has the perimeter equal to cross. So, we can say that cross has greater perimeter m comparison to square.