Construct a Quadrilateral 

When:

• Four sides and one diagonal are given.
• Two diagonals and three sides are given.
• Two adjacent sides and three angles are given.
• Three sides and two included angles are given.
• Other specific properties are known.

1. Construct quadrilaterals when four sides and one diagonal is given.

Construct a quadrilateral PQRS where PQ = 4 cm, Oq = 6 cm, RS = 5 cm, PS = 5.5 cm and PR= 7 cm

• Draw Δ PQR  using SSS construction condition.
• With P as the centre, draw an arc of radius 5.5 cm.
• With R as the centre, draw an arc of radius 5 cm.
• S is the point of intersection of the two arcs. Also, mark S and complete PQRS.
• PQRS is the required quadrilateral.

2. Construct a quadrilateral when two diagonals and  three sides  are given

Construct a quadrilateral ABCD, given that BC = 4.5 cm, AD = 5.5 cm, CD = 5 cm the diagonal AC = 5.5 cm and diagonal BD = 7 cm.

1. Draw ACD using SSS construction.
2. With D as centre, draw an arc of radius 7 cm.
3. With C as centre, draw an arc of radius 4.5 cm
4. Since B lies on both the arcs, B is the point intersection of the two arcs. Mark B and complete ABCD.

ABCD is the required quadrilateral

3. Construct a quadrilateral when two adjacent sides and three angles are given

Construct a quadrilateral ABCD, where sides adjacent sides AB & BC = 5 cm & 3 cm respectively. Also three angles the vertices A, B & C are 120 degree, 110 degree and 90 degree respectively.

Solution:

As per given question:
AB = 5 cm
BC = 3 cm
Angle A = 120 degree
Angle B = 110 degree
Angle C = 90 degree

Step of construction are as follows:

Step 1: Draw as line segment AB of 5 cm (as shown below)

Step 2: Use protractor and draw angle of 120 degree at vertex A (as shown below):

Step 3: Use protractor and draw angle of 110 degree at vertex B (as shown below):

Step 4: Use compass and 3 cm wide open. With B as center, draw an arc with cuts the arm of 110 degree angle; drawn in step 3. And mark point of intersection as C (as shown below)

Step 5: Use protractor and draw angle of 90 degree at point C (as shown below):

Step 6: You can observe that arm of 90 degree angle cuts the arm of angle 120 degree angle. Mark point of this intersection as D. And we get resultant Quadrilateral ABCD (as shown below):

4. Construct quadrilaterals when three sides and two included angles are given.

Construct a quadrilateral ABCD, where sides adjacent sides AB, BC, AD = 5 cm, 3 cm & 4 cm respectively. Also two included angles - Angle A and Angle B are 120 degree & 110 degree respectively.

Solution: As per given question:
AB = 5 cm
BC = 3 cm
AD = 4 cm
Angle A = 120 degree
Angle B = 110 degree

Step of construction are as follows:

Step 1: Draw as line segment AB of 5 cm (as shown below)

Step 2: Use protractor and draw angle of 120 degree at vertex A (as shown below):

Step 3: Use compass and 4 cm wide open. With A as center, draw an arc with cuts the arm of 120 degree angle; drawn in step 4. And mark point of intersection as D (as shown below)

Step 4: Use protractor and draw angle of 110 degree at vertex B (as shown below):

Step 5: Use compass and 3 cm wide open. With B as center, draw an arc with cuts the arm of 110 degree angle; drawn in step 3. And mark point of intersection as C (as shown below)

Step 6: Join point C & D and we get the resultant quadrilateral ABCD (as shown below):

3. Construct a rectangle whose diagonal = 5.8 cm and angle between the diagonals is 50∘ .