Question

Shinji is considering two different layouts for a new garden. He needs to put a length of fence around whichever layout he chooses. The following diagram shows both layouts on a coordinate grid. Which layout will need less length of fencing?

Answer 1

Layout B

Step-by-step explanation:

Since Shinji needs to put a length of fence around whichever layout he chooses, we find the perimeter of each of the figures.

Figure A

The vertices are P(-9,8), Q(-4,9), R(-1,-6) and S(-6,-7).

We use the distance formula to find the length of each of the sides.

`Distance=√((x_2-x_1)^2+(y_2-y_1)^2)`
`\begin{gathered} PQ=\sqrt[]{(-9-(-4))^2+(8_{}-9_{})^2}=\sqrt[]{(-9+4)^2+(-1)^2} \\ =\sqrt[]{(-5)^2+(-1)^2} \\ =\sqrt[]{26} \end{gathered}`

Similarly:

`\begin{gathered} PS=\sqrt[]{(-9-(-6))^2+(8_{}-(-7)_{})^2}=\sqrt[]{(-9+6)^2+(8+7)^2} \\ =\sqrt[]{(-3)^2+(15)^2} \\ =\sqrt[]{234} \end{gathered}`

Therefore, the perimeter of Figure A will be:

`\begin{gathered} =2\sqrt[]{26}+2\sqrt[]{234} \\ =40.79\text{ units} \end{gathered}`

Figure B

• 9-2 = 7 units

,

• 8-(-3)=8+3 = 11 units

The dimension of Figure B is 7 units by 11 units.

Therefore, the perimeter of Figure B is:

`\begin{gathered} =2(7+11) \\ =2*18 \\ =36\text{ units} \end{gathered}`

The layout in Figure B has a lower perimeter, therefore it will need less fencing.