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.

Perimeter and Area | IT

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

Understand the Question

To compare the areas of the blue rectangle (square) and the yellow triangle, find their areas in terms of side length $s$ .
Area of Square/Rectangle: $\text{Side} \times \text{Side} = s \times s = s^2$
Area of Triangle: $\dfrac{1}{2} \times \text{base} \times \text{height} = \dfrac{1}{2} \times 2s \times s = s^2$
Since both formulas give $s^2$ , their areas are equal.

(i) Is the area of the blue rectangle more or less than the area of the yellow triangle? Or is it the same? Why?

Step 1 · Calculate Area of the Blue Rectangle

Let the side of the blue square/rectangle be $s$ units. Diagram 1

\begin{aligned} \text{Area of blue rectangle} &= s \times s \\ &= s^2 \text{ square units} \end{aligned}

Step 2 · Calculate Area of the Yellow Triangle

The height of the yellow triangle is $s$ units and its base is $2s$ units. Diagram 2

\begin{aligned} \text{Area of yellow triangle} &= \dfrac{1}{2} \times \text{base} \times \text{height} \\[0.6em] &= \dfrac{1}{2} \times (2s) \times s \\[0.6em] &= s^2 \text{ square units} \end{aligned}

Answer

(i) The area of the blue rectangle and the yellow triangle are the same ( $s^2\text{ square units}$ ).

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

Step 1 · Find the Relationship Between the Areas

Both shapes have the exact same area: $\text{Area of blue rectangle} = \text{Area of yellow triangle} = s^2$

Additionally, a diagonal divides the blue rectangle into two identical triangles of area $\dfrac{s^2}{2}$ each. Therefore: $\text{Area of rectangle} = 2 \times \text{Area of one internal triangle}$

Answer

(ii) $\text{Area of Blue Rectangle} = \text{Area of Yellow Triangle} = 2 \times \text{Area of each half-triangle}$

Common Mistakes

Forgetting the Half Factor: Omitting $\dfrac{1}{2}$ in the triangle area formula $\dfrac{1}{2} \times \text{base} \times \text{height}$ , leading to an incorrect double area.
Misidentifying Dimensions: Overlooking that the base of the yellow triangle spans $2s$ while its height is only $s$ .

More questions in IT

Akshi says that the perimeter of this triangle shape is 9 units. Toshi says it can’t be 9 units and the perimeter will be more than 9 units. What do you think?

Write the perimeters of the figures below in terms of straight and diagonal units.

What is a similarity between a square and an equilateral triangle?

Split and rejoin

A rectangular paper chit of dimension $6\text{ cm} \times 4\text{ cm}$ is cut as shown into two equal pieces. These two pieces are joined in different ways.

Find out the length of the boundary (i.e., the perimeter) of each of the other arrangements below.

Arrange the two pieces to form a figure with a perimeter of 22 cm.

In previous grades, we arrived at the formula for the area of a rectangle and a square using square grid paper. Do you remember?

Look at the figures below and guess which one of them has a larger area.

Find the area of the following figures.

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.

Q10

Use your understanding from previous grades to calculate the area of any closed figure using grid paper and—

Find the area of blue triangle BAD.

Q11

Use your understanding from previous grades to calculate the area of any closed figure using grid paper and—

Find the area of red triangle $\text{ABE}$ .

Q12

Area of rectangle $\text{ABCD} = \text{\_\_\_\_\_\_\_\_}$

Q13

Using 9 unit squares, solve the following.

What is the smallest perimeter possible?
What is the largest perimeter possible?
Make a figure with a perimeter of 18 units.
Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning?

Q14

Let's do something tricky now! We have a figure below having perimeter 24 units.

Without calculating all over again, observe, think and find out what will be the change in the perimeter if a new square is attached as shown on the right.

Q15

Experiment placing this new square at different places and think what the change in perimeter will be. Can you place the square so that the perimeter: a) increases; b) decreases; c) stays the same?

Q16

Below is the house plan of Charan. It is in a rectangular plot. Look at the plan. What do you notice?

Some of the measurements are given.

a. Find the missing measurements.

b. Find out the area of his house.

Q17

Now, find out the missing dimensions and area of Sharan's home. Below is the plan:

Some of the measurements are given.

a. Find the missing measurements.

b. Find out the area of his house.

What are the dimensions of all the different rooms in Sharan's house? Compare the areas and perimeters of Sharan's house and Charan's house.

Q18

Area Maze Puzzles

In each figure, find the missing value of either the length of a side or the area of a region.