Constructing a Quadrilateral

Given below are various ways through which a quadrilateral can be constructed.

1. When length of 4 sides and a diagonal are given.

Step No.	Diagram	Details
1		Create a triangle PQR based on SSS construction condition according to the lengths of the sides and of the diagonal.
2		Make an arc (Point S) which is at a given distance away from P.
3		Make an arc on the earlier arc on S which is at a given distance from point R. Name the intersection of the two points as S.
4		Join S with P and R. This will complete the quadrilateral PQRS.

Problem: Construct a quadrilateral ABCD with following measurements.

AB = 4.5 cm, BC = 5.5 cm, CD = 4cm, AD = 6cm, AC = 7cm

Solution:

Step1: Draw Side BC = 5.5 cm and cut arcs above it from B (4.5 cm) and C (7 cm). Mark the intersection as A. Join AB and AC.

Step2: Draw and arc from A equal to 6 cm which is the length of AD.

Step3: Draw and arc from C equal to 4 cm which is the length of CD. Mark the intersection as D and join AD and CD.

2. When two diagonals and three sides are given.

Step No.	Diagram	Details
1		Create a triangle ACD based on SSS construction condition according to the lengths of the sides and of the diagonal.
2		Make an arc(with D as center) which is the length of the other diagonal.
3		Make an arc on the earlier arc which is at a given distance from point C (third side). Name the intersection of the two points as B.
4		Join B with A, C and D. This will complete the quadrilateral PQRS with the two diagonals.

Problem: Construct a quadrilateral GOLD with following measurements.

OL = 7.5 cm, GL = 6 cm, GD = 6 cm, LD = 5 cm, OD = 10 cm

 Solution:

Step1: Draw Side GD = 6 cm and cut arcs above it from G (6 cm) and D (5 cm). Mark the intersection as L. Join GL and DL

Step2: Draw and arc from L equal to 7.5 cm and from D equal to 10 cm which is the length of OL and OD resp

Step3: Mark the intersection as O and join OG, OL and OD.

When two adjacent sides and three angles are given.

Step No.	Diagram	Details
1		Draw one side and make an angle which is given.
2		Make another given angle from the newly formed side (X-axis).
3		Make another given angle. This would intersect with the Y axis formed above. The quadrilateral will be formed.

Problem: Construct a quadrilateral PLAN with following measurements.

PL = 4 cm, LA = 6.5 cm, ∠P= 90°, ∠A = 110°, ∠N = 85°

Solution:

The sum of the angles of a quadrilateral is 360°. So, ∠P + ∠L + ∠A + ∠N = 360°.

90° + ∠L + 110° + 85° = 360°

∠L = 75°

Step1: Draw Side PL = 4 cm and draw an angle of 75° at point L. As vertex A is 6.5 cm away from L, cut a line segment LA of 6.5 cm from this ray.

Step2: Again draw an angle of 110° at point A.

Step3: draw an angle of 90° at point P. This ray will meet the previously drawn ray from A at point N.

When three sides and two included angles are given.

Step No.	Diagram	Details
1		Draw one side and make an angle which is given.
2		Make another given angle from the existing side (BC).
3		Mark the point on the new side (Y axis) at the point equal to the length of the given side. Mark it as D.
4		Join the points AD to form the fourth side of the quadrilateral. The quadrilateral is now complete.

Problem: Construct a quadrilateral TRUE with following measurements.

TR = 3.5 cm, RU = 3 cm, UE = 4 cm, ∠R = 75°, ∠U = 120°

 Solution:

Step1: Draw Side RU = 3 cm and an angle of 120° at point U. As vertex E is 4 cm away from U, cut a line segment UE of 4 cm from this ray.

Step2: Draw an angle of 75° at point R. As vertex T is 3.5 cm away from vertex R, cut a line segment RT of 3.5 cm from this ray.

Step3: Join T to E.