Now, see the figures below. Is the area of the blue rectangle more or less than the area of the yellow triangle? Or is it the same? Why? Can you see some relationship between the blue rectangle and the yellow triangle and their areas? Write the relationship here.

The area of the blue rectangle and the yellow triangle are the same. Step 1 — Area of the blue rectangle Let the side of the blue square be units. The blue shape is a square. Its area is found by multiplying its side by itself. Area of blue rectangle = Step 2 — Area of the yellow triangle Look at the yellow triangle. Its height is the same as the side of the blue square. So, its height is units. Its base is twice the side of the blue square. So, its base is units. The area of a triangle is half of its base multiplied by its height. Area of yellow triangle = Step 3 — Comparing areas and finding relationships We see that the area of the blue rectangle is square units. We also see that the area of the yellow triangle is square units. So, their areas are equal. Now, let us look at the blue rectangle again. A diagonal line cuts the blue rectangle into two triangles. These two triangles are exactly the same size. So, the area of the whole blue rectangle is twice the area of one of these triangles. Answer > (i) The area of the blue rectangle and the yellow triangle are equal. > (ii) The areas are the same because if the side of the blue square is units, its area is square units. The yellow triangle has a height of units and a base of units, so its area is square units. > (iii) Area of Rectangle = 2 × Area of triangle

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Question 9

Now, see the figures below. Is the area of the blue rectangle more or less than the area of the yellow triangle? Or is it the same? Why?

Can you see some relationship between the blue rectangle and the yellow triangle and their areas? Write the relationship here.

Solution

The area of the blue rectangle and the yellow triangle are the same.

Step 1 — Area of the blue rectangle Let the side of the blue square be $s$ units. The blue shape is a square. Its area is found by multiplying its side by itself. Area of blue rectangle = $s \times s$ $= s^2 \text{ square units}$

Diagram 1

Step 2 — Area of the yellow triangle Look at the yellow triangle. Its height is the same as the side of the blue square. So, its height is $s$ units. Its base is twice the side of the blue square. So, its base is $2s$ units. The area of a triangle is half of its base multiplied by its height. Area of yellow triangle = $(1/2) \times \text{base} \times \text{height}$ $= (1/2) \times (2s) \times s$ $= s^2 \text{ square units}$

Diagram 2

Step 3 — Comparing areas and finding relationships We see that the area of the blue rectangle is $s^2$ square units. We also see that the area of the yellow triangle is $s^2$ square units. So, their areas are equal. Now, let us look at the blue rectangle again. A diagonal line cuts the blue rectangle into two triangles. These two triangles are exactly the same size. So, the area of the whole blue rectangle is twice the area of one of these triangles.

Answer

(i) The area of the blue rectangle and the yellow triangle are equal. (ii) The areas are the same because if the side of the blue square is $s$ units, its area is $s^2$ square units. The yellow triangle has a height of $s$ units and a base of $2s$ units, so its area is $(1/2) \times (2s) \times s = s^2$ square units. (iii) Area of Rectangle = 2 × Area of triangle

Question 9

Answer

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