Question 10

We can find the area of a triangle by multiplying half of its base by its height.

Step 1 — Find the base and height of triangle BAD

Let us look at the blue triangle BAD. We can choose the side AB as the base. Let us count the grid units along AB. The length of AB is 5 units. The height is the perpendicular distance from point D to the base AB. This height is the length of the side DA. Let us count the grid units along DA. The length of DA is 4 units.

Step 2 — Calculate the area of blue triangle BAD

The formula for the area of a triangle is half times base times height. Area of triangle BAD = $1/2 \times \text{base} \times \text{height}$ $= 1/2 \times 5 \text{ units} \times 4 \text{ units}$ $= 1/2 \times 20 \text{ sq units}$

$\boxed{\text{Area of blue triangle BAD} = 10 \text{ sq units}}$

Step 3 — Find the base and height of triangle ABE

Now let us look at the red triangle ABE. We can choose the side AB as the base. Let us count the grid units along AB. The length of AB is 5 units. The height is the perpendicular distance from point E to the base AB. This height is the length of the side EF. Let us count the grid units along EF. The length of EF is 4 units.

Step 4 — Calculate the area of red triangle ABE

The formula for the area of a triangle is half times base times height. Area of triangle ABE = $1/2 \times \text{base} \times \text{height}$ $= 1/2 \times 5 \text{ units} \times 4 \text{ units}$ $= 1/2 \times 20 \text{ sq units}$

$\boxed{\text{Area of red triangle ABE} = 10 \text{ sq units}}$

Step 5 — Calculate the area of rectangle ABCD

Let us look at the rectangle ABCD. The length of the rectangle is AB. Let us count the grid units along AB. The length of AB is 5 units. The width of the rectangle is AD. Let us count the grid units along AD. The width of AD is 4 units. The formula for the area of a rectangle is length times width. Area of rectangle ABCD = $\text{length} \times \text{width}$ $= 5 \text{ units} \times 4 \text{ units}$

$\boxed{\text{Area of rectangle ABCD} = 20 \text{ sq units}}$

Step 6 — Draw a conclusion

We found the area of blue triangle BAD. It is 10 sq units. We found the area of red triangle ABE. It is 10 sq units. We found the area of rectangle ABCD. It is 20 sq units. Both triangles BAD and ABE share the same base AB. Their heights are also the same, 4 units. The rectangle ABCD also has base AB and height AD (or BC). The area of each triangle is half the area of the rectangle.

$\boxed{\text{When a triangle and a rectangle have the same base and the same height, the triangle's area is half of the rectangle's area.}}$

Answer

(i) The area of blue triangle BAD = 10 sq units (ii) The area of red triangle ABE = 10 sq units (iii) The area of rectangle ABCD = 20 sq units. When a triangle and a rectangle have the same base and the same height, the triangle's area is half of the rectangle's area.