Question 1

We need to find the total length around the triangle shape.

Step 1 — Find lengths of straight sides

Let us look at the grid of dots. The distance between two nearby dots is 1 unit. This is true for horizontal or vertical steps. The first red side goes straight up. It connects dots from the bottom row to the fourth row. This covers 3 unit distances. So, the first red side is 3 units long.

The second red side goes straight across. It connects dots from the first column to the fourth column. This covers 3 unit distances. So, the second red side is 3 units long.

$\text{Length of vertical side} = 3 \text{ units}$ $\text{Length of horizontal side} = 3 \text{ units}$

Step 2 — Find length of diagonal side

Let us look at the blue diagonal side. This side connects two dots. One dot is at the bottom-left. The other dot is at the top-right. We move right 3 units. We also move up 3 units. A diagonal line is always longer than a straight line. It is longer than 3 units. A square's diagonal is always longer than its side. So, the diagonal side is longer than 3 units.

We can find its exact length. We use a special method. Take the horizontal distance, which is 3. Multiply it by itself: $3 \times 3 = 9$. Take the vertical distance, which is 3. Multiply it by itself: $3 \times 3 = 9$. Add these two numbers: $9 + 9 = 18$. We need a number. Multiply it by itself. It must give 18. This number is about 4.24. So, the diagonal side is about 4.24 units long.

$\text{Length of diagonal side} \approx 4.24 \text{ units}$

Step 3 — Calculate the total perimeter

The perimeter is the total length of all sides. We add the lengths of the three sides. Perimeter = vertical side + horizontal side + diagonal side.

$\text{Perimeter} = 3 \text{ units} + 3 \text{ units} + 4.24 \text{ units}$ $\text{Perimeter} = 6 \text{ units} + 4.24 \text{ units}$ $\text{Perimeter} = 10.24 \text{ units}$

$\boxed{\text{Perimeter} \approx 10.24 \text{ units}}$

Step 4 — Compare with Akshi and Toshi

Akshi says the perimeter is 9 units. Our calculated perimeter is about 10.24 units. So, Akshi is not correct.

Toshi says the perimeter is more than 9 units. Our calculated perimeter is about 10.24 units. This is more than 9 units. So, Toshi is correct. The diagonal line is longer than 3 units. This makes the total perimeter more than 9 units.

Answer

(i) Akshi's statement that the perimeter is 9 units is not correct. (ii) Toshi is right. (iii) The perimeter is more than 9 units. This is because the diagonal line is longer than 3 units. The correct perimeter is approximately 10.24 units. This is more than 10 units.