**Constructing a Quadrilateral:**

**1. Steps to construct a quadrilateral when lengths of four sides and a diagonal are given
**Let us understand it by an example:

** Example 1:** Construct a quadrilateral ABCD such that AB = 5 cm, BC = 7 cm, CD = 6 cm, DA = 6.5 cm and AC = 8 cm.

Step 1: Construct Î”ABC using SSS criterion.Step 2: Now, to draw the fourth point D. Keep A as center and draw and arc of radius 6.5 cm on the opposite side of point B.Step 3: Draw an another arc keeping center as C of radius 6 cm, such that it intersects the arc drawn in above step.Step 4: The intersecting point of both the arcs will be point D. Draw the lines from point A and C to complete the quadrilateral ABCD.

** Example 2:** Construct a parallelogram MORE with OR = 6cm, RE = 4.5 cm and EO = 7.5cm.

Step 1: Construct Î”EOR using SSS criterionStep 2: To draw point M, take O and E as centers with arcs of radius 4.5 cm and 6 cm respectively. These two must intersect and this is the point M.Step 3: To complete the parallelogram MORE, join O and E to M.

**2. Steps to construct a quadrilateral when two diagonals and three sides are given
**Let us understand it by an example:

** Example 1:** Construct a quadrilateral ABCD such that the diagonals AC = 6.5 cm and BD = 8 cm. The measure of the other sides will be BC = 5.5 cm, AD = 6.5 cm and CD = 6 cm.

Step 1: Construct Î”ACD using SSS criterion.Step 2: Now, to draw the fourth point B. Keep D as center and draw and arc of radius 8 cm on the opposite side of point D.Step 3: Draw an another arc keeping center as C of radius 5.5 cm, such that it intersects the arc drawn in above step.Step 4: The intersecting point of both the arcs will be point B. Draw the lines from point A and C to complete the quadrilateral ABCD.

Step 1: Draw a line segment BN of length 6.5 cm. Now, draw perpendicular bisectors of BN with the intersection point as O.Step 2: Take O as center and draw arcs of length 3.25 cm which intersects the perpendicular bisector at point D and E.Step 3: To complete the rhombus BEND, join D to B and E to N.

**3. Steps to construct a quadrilateral when two adjacent sides and three angles are given
**Let us understand it by an example:

** Example 1:** Construct a quadrilateral ABCD such that the AB = 4.5 cm and BC = 7.5 cm. The measure of the angles are âˆ A = 75áµ’, âˆ B = 105áµ’ and âˆ C = 120áµ’.

Step 1: Draw AB = 4.5 cm. Then make âˆ B = 105áµ’ and draw BC = 7.5 cm.Step 2: Make âˆ C = 120áµ’.Step 3: Make âˆ A = 75áµ’ such that it intersects the line extended from point C. At last, complete the quadrilateral ABCD.

So, âˆ P + âˆ L + âˆ A + âˆ N = 360áµ’.

90áµ’ + âˆ L + 110áµ’ + 85áµ’ = 360áµ’

Thus, âˆ L = 75áµ’

Step 1: Draw line segment PL of length 4 cm. Now, draw an angle of 75áµ’ at point L; cut a line segment LA of 6.5 cm from point L.Step 2: From point A, draw an angle of 110áµ’.Step 3: From point P, draw an angle of 90. This ray will meet the ray from point A intersecting at point N.

**3. Steps to construct a quadrilateral when three sides and two angles are given
**Let us understand it by an example:

** Example 1:** Construct a quadrilateral ABCD such that the AB = 5 cm, BC = 6 cm and CD = 7.5 cm. The measure of the angles are âˆ B = 105áµ’ and âˆ C = 80áµ’.

Step 1: Draw line BC = 6cm. Now, draw âˆ CBA = 105áµ’ and AB = 5cm.Step 2: Draw âˆ C = 80áµ’ towards point B.Step 3: Draw an arc of length 7.5 cm from point C, since, it is given CD = 7.5 cm.Step 4: Join the line AD and complete the quadrilateral ABCD.

** Example 2:** Draw a quadrilateral TRUE with TR = 3.5 cm, RU = 3 cm, UE = 4 cm, âˆ R = 75áµ’ and âˆ U = 120áµ’.

Step 1: Draw line segment RU of 3 cm. Now, make angle âˆ U = 120áµ’. Keeping U as centre cut a line segment UE of 4 cm.Step 2: Draw âˆ R = 75áµ’. And keeping R as centre cut a line segment RT of 3.5 cm.Step 3: To complete quadrilateral TRUE, join T to E.

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