## Construction of a Quadrilateral

It is very easy to construct a quadrilateral when its five measurements are determined that is

- The length of the four sides and the length of its diagonal is known
- The length of the three sides and the length of the two diagonals are known
- If the three angles and two adjacent sides are given
- If the three sides and two angles are given

## 4 Sides and 1 Diagonal

### Construction of a Quadrilateral when different measures of sides and angles are given

A **unique quadrilateral** can be constructed when the following measurements are given:

**Four sides**and**one diagonal**.**Two diagonals**and**three sides**.**Two adjacent sides**and**three angles**.**Three sides**and two**included angles**.- When other special properties are known.

### SSS Construction

To construct a △ABC, the length of whose sides are, AB = *x* cm, BC = *y* cm, and AC = *z* cm, we will do it in the following manner:

- Construct a line segment AB, whose length is x cm.
- With A as the center, draw an arc of radius z cm.
- With B as the center, draw an arc of radius y cm on the same side. The point where the arcs intersect is the required point C.
- Join AC and BC.

△ ABC is the required triangle.

### Construction of a Quadrilateral when four sides and one diagonal are given

Suppose we have to construct a quadrilateral PQRS, where PQ = 4 cm, QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR = 7 cm.

Step 1: Draw a rough sketch to visualize the quadrilateral.

Step 2: Draw △ PQR as it can be constructed using SSS construction condition.

Step 3: Now we have to locate S, which is at a distance of 5.5 cm from P and 5 cm from R. Also it will be on the opposite side of Q.

With P as center draw an arc of radius 5.5 cm. With R as center draw an arc of radius 5 cm.

S is the point of intersection of the two arcs.

Step 4: Join PS and RS. PQRS is the required quadrilateral.

## 3 Sides and 2 Diagonals

### Construction of a Quadrilateral when two diagonals and three sides are given

Construct a quadrilateral ABCD given, AB = 7 cm, AD = 6 cm, AC = 7 cm, BD = 7.5 cm and BC = 4 cm.

[Make a rough figure for your reference]

Steps of construction of the quadrilateral:

Step 1: △ABC can be drawn by SSS construction condition since all its sides are known.

Step 2: With A as center and radius 6 cm (AD), draw an arc.

Step 3: With B as center and radius 7.5 cm (BD) draw another arc to cut the previous arc at D

Step 4: Join AD, BD, and CD.

ABCD is the required quadrilateral

## 2 Adjacent Sides and 3 Angles

### Construction of a Quadrilateral when two adjacent sides and three angles are given

Construct a quadrilateral ALPN, where AL = 6.5 cm, LP = 4 cm, ∠NAL = 110∘, ∠ALP = 75∘ and ∠LPN = 90∘.

[Draw a rough Sketch for your reference]:

Steps of construction of the quadrilateral:

Step 1: Draw the line segment AL of length 6.5 cm.

Step 2: Make ∠ALY = 75 ∘ at L.

Step 3: Make ∠LAX = 110∘ at A.

Step 4: With L as center and radius equal to 4 cm, cut an arc on the ray LY at P.

Step 5: Make ∠LPZ = 90∘ at P.

Step 6: Name the point of intersection of rays PZ and AX as N.

ALPN is the required quadrilateral.

## 3 Sides and 2 Included Angles

### Construction of a Quadrilateral when three Sides and two included angles are given

Construct a quadrilateral ABCD, Where AB = 4.5 cm; BC = 3.5 cm, CD = 5 cm ∠ABC = 45∘, ∠BCD = 150∘

[Make a rough figure for your reference]

Steps of construction of the quadrilateral:

Step 1: Draw a line segment BC of length 3.5 cm.

Step 2: Make ∠LBC = 45∘.

Step 3: Make ∠BCM = 150∘.

Step 4: With B as center and radius equal to 4.5 cm, cut an arc on the ray LB at A.

Step 5: With C as the center and radius equal to 5 cm, cut an arc on the ray CM at D.

Step 6: Join AD.

ABCD is the required quadrilateral.

## Special Quadrilaterals

### Construction of a Quadrilateral When Other Special Properties Are Known

Construct a rhombus PQRS with diagonals PR = 5.2 cm and QS = 6.4 cm

[Make a rough figure for your reference]

Note: Diagonals of a rhombus are perpendicular bisectors of each other.

Steps of construction of the Rhombus:

Step 1: Draw a line segment PR of length 5.2 cm.

Step 2: Draw the perpendicular bisector of PR. Name the point O, where the perpendicular bisector of PR and PR intersect.

Step 3: With O as center and radius equal to 3.2 cm cut arcs on both sides of the perpendicular bisector. Name them as Q and S.

Step 4: Join, PQ, QR, RS, and PS.

PQRS is the required rhombus.

## Introduction to Practical Geometry

### Number of measurements necessary for construction of a unique Quadrilateral

To draw a **unique quadrilateral **we need at least **five measurements** of sides and angles**. **However, it is not necessary that we will get a unique quadrilateral if we have the measurements of **any five combinations** of sides and angles.

For example, a unique quadrilateral can be drawn if we are given the measurement of four sides and one diagonal of a quadrilateral.

However, a unique quadrilateral will not be drawn if we are given the measurement of two diagonals and three angles of a quadrilateral.