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Construction of Quadrilaterals – (Methods, Steps & Examples)

construction of quadrilaterals

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A quadrilateral is a closed shape and a type of polygon that has four sides, four vertices, and four angles. It is formed by joining four non-collinear points. The sum of four angles in a quadrilateral is $360^{\circ}$.

construction of quadrilaterals

Let’s understand the construction of quadrilaterals with steps and examples.

Construction of Quadrilaterals

A quadrilateral is a closed 2D geometrical shape with four sides and four angles. The sum of four angles of a quadrilateral is $360^{\circ}$. There are three methods of constructing a quadrilateral depending on the parameter known for construction. You can construct a unique quadrilateral when

  • four sides and one diagonal is given
  • three angles and two included sides are given
  • three sides and two included angles are given

Note: To construct a unique quadrilateral, at least five parameters are required.

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Construction of Quadrilaterals When Four Sides and One Diagonal is Given

Let’s consider an example of a quadrilateral to understand the construction of quadrilaterals when four sides and one diagonal is given.

Construct a quadrilateral $\text{ABCD}$ with $\text{BC} = 4.5 \text{ cm}$, $\text{DA} = 5.5 \text{ cm}$,  $\text{CD} = 5 \text{ cm}$ and the diagonal $\text{AC} = 5.5 \text{ cm}$, diagonal $\text{BD} = 7 \text{ cm}$.

The following are the steps to construct a unique quadrilateral with given four sides and one diagonal.

Step 1: Construct a $\triangle \text{ACD}$ using $\text{SSS}$ construction criteria.

construction of quadrilaterals

Read the construction of triangles to understand the steps.

Step 2: Taking $\text{D}$ as a centre and radius equal to the length of diagonal $\text{BD} = 7 \text{ cm}$ draw an arc opposite side of the diagonal $\text{AC}$

construction of quadrilaterals

Step 3: Taking $\text{C}$ as a centre and radius equal to the length of side $\text{BC} = 4.5 \text{ cm}$ draw an arc intersecting the arc drawn in Step 2 at $\text{B}$

construction of quadrilaterals

Step 4: Join the points $\text{A}$, $\text{B}$ and $\text{C}$, $\text{B}$

construction of quadrilaterals

$\text{ABCD}$ is the required quadrilateral.

Construction of Quadrilaterals When Three Sides and Two Included Angles are Given

Let’s consider an example of a quadrilateral to understand the construction of quadrilaterals when three sides and two included angles are given.

Construct a quadrilateral $\text{PQRS}$, with sides $\text{PQ} = 3.5 \text{ cm}$, $\text{QR} = 4.5 \text{ cm}$, $\text{RS} = 4 \text{ cm}$ and $\angle {\text{Q}} = 110^{\circ}$, $\angle {\text{R}} = 80^{\circ}$.

The following are the steps to construct a unique quadrilateral with given three angles and two included sides. 
Step 1: Draw the side $\text{QR} = 4.5 \text{ cm}$

construction of quadrilaterals

Step 2: At point $\text{Q}$ with base line as $\text{QR}$, draw an angle of $110^{\circ}$

construction of quadrilaterals

Step 3: At point $\text{R}$ with base line as $\text{QR}$, draw an angle of $80^{\circ}$

construction of quadrilaterals

Step 4: With $\text{Q}$ centre and radius equal to side $\text{PQ} = 3.5 \text{ cm}$, draw an arc intersecting the line at $\text{Q}$ at the point $\text{P}$.

construction of quadrilaterals

Step 5: With $\text{R}$ centre and radius equal to side $\text{RS} = 4 \text{ cm}$, draw an arc intersecting the line at $\text{R}$ at the point $\text{S}$.

construction of quadrilaterals

Step 6: Join the points $\text{P}$ and $\text{S}$

construction of quadrilaterals

$\text{PQRS}$ is the required quadrilateral.

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Construction of Quadrilaterals When Three Angles and Two Included Sides are Given

Let’s consider an example of a quadrilateral to understand the construction of quadrilaterals when three angles and two included sides are given.

Construct a quadrilateral $\text{ABCD}$, with sides $\text{AB} = 4 \text{ cm}$, $\text{BC} = 5 \text{ cm}$, $\text{DA} = 3 \text{ cm}$ and $\angle \text{{A}} = 100^{\circ}$, $\angle \text{{B}} = 120^{\circ}$.

The following are the steps to construct a unique quadrilateral with given three angles and two included sides.

Step 1: Draw the side $\text{AB} = 5 \text{ cm}$

construction of quadrilaterals

Step 2: At the point $\text{A}$, draw an angle of $120^{\circ}$, with $\text{AB}$ as a base

construction of quadrilaterals

Step 3: At the point $\text{B}$, draw an angle of $110^{\circ}$, with $\text{BA}$ as a base

construction of quadrilaterals

Step 4: With $\text{B}$ as a centre and radius equal to $3 \text{cm}$, draw an arc intersecting the line at $\text{B}$, at $\text{C}$

construction of quadrilaterals

Step 5: At the point $\text{C}$, draw an angle of $90^{\circ}$, with $\text{BC}$ as a base intersecting at $\text{D}$

construction of quadrilaterals

$\text{ABCD}$ is the required quadrilateral.

Practice Problems

  1. Construct a quadrilateral $\text{LMNO}$ in which $\text{LM} = 4.2 \text{ cm}$, $\text{MN} = 6 \text{ cm}$, $\text{NO} = 5.2 \text{ cm}$, $\text{OL} = 5 \text{ cm}$ and $\text{LN} = 8 \text{ cm}$. 
  2. Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = 3.5 \text{ cm}$, $\text{BC} = 3.8 \text{ cm}$, $\text{CD} = \text{DA} = 4.5 \text{ cm}$ and diagonal $\text{BD} = 5.6 \text{ cm}$. 
  3. Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = 3.6 \text{ cm}$, $\text{BC} = 3.3 \text{ cm}$, $\text{AD} = 2.7 \text{ cm}$, diagonal $\text{AC} = 4.6 \text{ cm}$ and diagonal $\text{BD} = 4 \text{ cm}$.
  4. Construct a quadrilateral $\text{LMNO}$ in which $\text{LN} = \text{LO} = 6 \text{ cm}$, $\text{MN} = 7.5 \text{ cm}$ , $\text{MO} = 10 \text{ cm}$ and $\text{NO} = 5 \text{ cm}$. 
  5. Construct a quadrilateral $\text{PQRS}$ in which $\text{PQ} = 5.4 \text{ cm}$, $\text{BC} = 6 \text{ cm}$, $\text{QR} = 4.6 \text{ cm}$, $\text{RS} = 4.3 \text{ cm}$, $\text{SP} = 3.5 \text{cm}$ and diagonal $\text{BD} = 5.6 \text{ cm}$.
  6. Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = \text{BC} = 3.5 \text{ cm}$, $\text{AD} = \text{CD} = 5.2 \text{ cm}$ and $\angle \text{ABC} = 120^{\circ}$.
  7. Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = 2.9 \text{ cm}$, $\text{BC} = 3.2 \text{ cm}$, $\text{CD} = 2.7 \text{ cm}$, $\text{DA} = 3.4 \text{ cm}$ and $\angle \text{A} = 70^{\circ}$.
  8. Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = 3.5 \text{ cm}$, $\text{BC} = 5 \text{ cm}$, $\text{CD} = 4.6 \text{ cm}$, $\angle \text{B} = 125^{\circ}$ and $\angle \text{C} = 60^{\circ}$.
  9. Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = 5.6 \text{ cm}$, $\text{BC} = 4 \text{ cm}$, $\angle \text{A} = 50^{\circ}$, $\angle \text{B} = 105^{\circ}$ and $\angle \text{D} = 80^{\circ}$.

FAQs

How many minimum measurements are required to construct a unique quadrilateral?

The minimum number of measurements required to construct a unique quadrilateral is five.

What are the different methods of construction of quadrilaterals?

There are three different methods of constructing a unique quadrilateral. These are when
a) four sides and one diagonal is given
b) three angles and two included sides are given
c) three sides and two included angles are given

Conclusion

A quadrilateral is a closed 2D geometrical shape with four sides and four angles. The sum of four angles of a quadrilateral is $360^{\circ}$. To construct a unique quadrilateral, at least five parameters are required. You can construct a unique quadrilateral when four sides and one diagonal is given, or three angles and two included sides are given or three sides and two included angles are given.

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