PDF chapter test

A rectangle can be constructed using two set of measurements.

The two instances are:
1. When the length and breadth is known.

2. When a side and a diagonal is known.
Method $$1$$: When the length and breadth is known
Let us construct a rectangle with length as $$10$$ $$cm$$ and breadth as $$4$$ $$cm$$. Let us also calculate the area of the rectangle thus constructed.

Step $$1$$: Draw a rough diagram using the measurements known.

Step $$2$$: Draw a line segment $$AB$$ of length $$10$$ $$cm$$.

Step $$3$$: With $$A$$ as centre, draw a perpendicular line.

Step $$4$$: With $$A$$ as centre and with $$4$$ $$cm$$ as radius, draw an arc on the perpendicular line and mark the intersection as $$D$$.

Step $$5$$: With $$D$$ as centre and with $$10$$ $$cm$$ as radius, draw an arc. Similarly, with $$B$$ as centre and with $$4$$ $$cm$$ as radius, cut the existing arc and mark the intersection as $$C$$.

Step $$6$$: Now join $$CD$$ and $$BC$$ to obtain the desired rectangle.

To find the area of the rectangle:

$$\text{Area of the rectangle} = \text{Length} \times \text{Breadth}$$

$$\text{Area of the rectangle} = l \times b$$

We know that, $$l = 10$$ $$cm$$, and $$b = 4$$ $$cm$$.

Now, $$\text{Area of the rectangle} = 10 \times 4$$

$$= 40$$ $$cm^2$$
Method $$2$$: When a side and a diagonal is known
Let us construct rectangle with a side as $$8$$ $$cm$$ and the diagonal as $$10$$ $$cm$$. Let us also calculate the area of the obtained rectangle.

Step $$1$$: Draw a rough diagram with the known measurements.

Step $$2$$: Draw a line segment $$AB$$ of length $$8$$ $$cm$$.

Step $$3$$: With $$A$$ as centre and with $$10$$ $$cm$$ as radius, draw an arc.

Step $$4$$: With $$B$$ as centre, draw a perpendicular line until it meets the arc already drawn. Mark the intersection as $$C$$.

Step $$5$$: Measure $$BC$$. In this case, $$BC = 6$$ $$cm$$, With $$A$$ as centre and with $$6$$ $$cm$$ as radius, draw an arc. Similarly, with $$C$$ as centre and with $$8$$ $$cm$$ as radius, cut the arc. Mark the intersection as $$D$$.

Step $$6$$: Join $$AD$$ and $$CD$$ to form the rectangle.

To find the area of the rectangle:

$$\text{Area of the rectangle} = \text{Length} \times \text{Breadth}$$

$$\text{Area of the rectangle} = l \times b$$

We know that, $$l = 8$$ $$cm$$, and $$b = 6$$ $$cm$$.

Now, $$\text{Area of the rectangle} = 8 \times 6$$

$$= 48$$ $$cm^2$$