 UPSKILL MATH PLUS

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For any quadrilateral to be constructed, we should atleast know partial measurements of the quadrilateral.

The four ways with which a rhombus could be constructed, include:
1. When one side and one diagonal is known.

2. When one side and one angle is known.

3. When two diagonals are known.

4. When one diagonal and one angle is known.
Method $$1$$: When one side and one diagonal is known
Let us try to construct a rhombus with one of its sides as $$6.5$$ $$cm$$ and one of its diagonals as $$12$$ $$cm$$. Let us also find the area of the constructed rhombus.

Step $$1$$: Draw a rough diagram with the known measurements. Step $$2$$: In a sheet of paper, draw a line segment $$AB$$ of $$6.5$$ $$cm$$ in length. Step $$3$$: With $$A$$ as centre and with $$12$$ $$cm$$ as the radius, draw an arc. With $$B$$ as centre and $$6.5$$ $$cm$$ as the radius, cut the arc and mark the intersection as $$C$$. Join $$AC$$ and $$BC$$. Step $$4$$: With $$A$$ as centre, and with $$6.5$$ $$cm$$ as radius, draw an arc. With $$C$$ as centre, and with $$6.5$$ $$cm$$ as radius, cut the existing arc. Mark the intersection as $$D$$. Join $$AD$$ and $$CD$$ to get the desired rhombus. To find the area of the rhombus:

$$\text{Area of a rhombus}$$ $$=$$ $$\frac{1}{2}$$ $$\times$$ $$d_1$$ $$\times$$ $$d_2$$

$$d_1$$ and $$d_2$$ are the diagonals of the rhombus.

It is now understood that the lengths of both the diagonals are required to measure the area of the rhombus.

We already know the length of one of the diagonals (say $$d_1$$). It is $$12$$ $$cm$$.

To know the measurement of the other diagonal (that is, $$d_2$$), we should join $$BD$$ and measure the length manually.

Say the length of the other diagonal ($$d_2$$) is $$5$$ $$cm$$. After substituting the known values, the formula to find the area of the rhombus becomes:

$$\text{Area of the rhombus}$$ $$=$$ $$\frac{1}{2}$$ $$\times$$ $$12$$ $$\times$$ $$5$$

$$= 30$$ $$cm^2$$
Method $$2$$: When one side and one angle is given
Let us construct a rhombus $$ABCD$$ with $$AB$$ as $$13 cm$$ and $$\angle A$$ as $$75^\circ$$. Let us also find the area of the constructed rhombus.

Step $$1$$: Draw a rough diagram with the known measurements. Step $$2$$: In a sheet of paper, draw a line segment $$AB$$ of $$13$$ $$cm$$ in length. Step $$3$$: With $$A$$ as centre, measure $$75^\circ$$ and draw a line. Step $$4$$: With $$A$$ as centre and with $$13$$ $$cm$$ as radius, draw an arc over the line. Step $$5$$: With $$D$$ as centre and with $$13$$ $$cm$$ as radius, draw an arc. Similarly with $$B$$ as radius and with $$13$$ $$cm$$ as radius, cut the existing arc. Mark the intersection as $$C$$. Step $$6$$: Now join $$CD$$ and $$BC$$ to form the desired rhombus. To find the area of the rhombus:

$$\text{Area of a rhombus}$$ $$=$$ $$\frac{1}{2}$$ $$\times$$ $$d_1$$ $$\times$$ $$d_2$$

Lengths of both the diagonals are required to measure the area of the rhombus.

To know the measurement of the diagonals, we should join $$AC$$ and $$BD$$ to measure the length manually.

On measuring the diagonals, the lengths of the diagonals are $$20.6$$ $$cm$$ and $$15.8$$ $$cm$$ respectively.

Now, the formula to find the area of the rhombus becomes:

$$\text{Area of the rhombus}$$ $$=$$ $$\frac{1}{2}$$ $$\times$$ $$20.8$$ $$\times$$ $$15.8$$

$$= 164.3$$ $$cm^2$$