SPECIAL CASES QUADRILTERALS

CONSTRUCTING A QUADRILATERAL WHEN FOUR SIDES AND A DIAGONAL OF A QUADRILATERAL ARE GIVEN:

We consider the quadrilateral ABCD as a figure made of two triangles:

• ΔABC and ΔADC when diagonal AC as the common side is given.
• ΔABD and ΔBCD when diagonal BD as the common side is given.

In order to draw the quadrilateral when four sides and one diagonal are given, we first draw a rough sketch of the quadrilateral and write its dimensions along the sides and then we divide it into two triangles which can be drawn conveniently.

STEPS:

Step 1: Construct a triangle ABD (say).

Step 2: Find point C opposite to the vertex A as follows. With B as the centre and given radius, draw an arc on the other side of BD. Similarly with D as the centre and given radius, draw another arc intersecting the previous arc. The point of intersection of these arcs is marked as C.

Step 3: join points B and C, and D and C.

question 1. Construct a quadrilateral ABCD, in which AB = 5.5 cm, AD = 4.4 cm, CD = 6.5 cm, and AC = 6.5 cm and BD = 7.1 cm.

Solution: First we draw a rough sketch of quadrilateral ABCD. It is evident from the rough sketch that we have sufficient data to draw triangles ADC and ABD.

• Join AD, CD, AB and CB to obtain the required quadrilateral.

question 2. Construct a quadrilateral ABCD, when AB = 4 cm, BC = 6.0 cm, CD = DA = 5.2 cm and AC = 8 cm.

Sol. (i) Construct AC = 8 cm as shown in figure.

Thus, ABCD is the required quadrilateral.

question 3. Construct the quadrilateral ABCD in which AB = 4.5 cm, BC = 5.5 cm, CD = 4, cm, AD = 6 cm and AC = 7 cm.

•  Repeat the construction to obtain ΔADC taking AD = 6 cm and DC = 4 cm.
•  ABCD is the required quadrilateral 

question 4. Construct the quadrilateral JUMP in which JU = 3.5 cm, UM = 4 cm, MP = 5 cm, PJ = 4.5 cm and PU = 4 cm.

question 5. Construct a parallelogram ABCD in which, AB = 6 cm, BC = 4.5 cm and diagonal AC = 6.8 cm.

•  Join BC and AC.
• With A as centre and radius 4.5 draw an arc.
• With C as centre and radius 6 cm draw another arc so as it cuts the previously drawn arc of step (v) at D. 
• Join DA and DC.

Then, ABCD is the required parallelogram.

CONSTRUCTING A QUADRILATERAL WHEN LENGTHS OF ITS THREE SIDES AND TWO INCLUDED ANGLES:

STEPS:
Step 1: Draw a line segment MN (say) of given length.

Step 2: Construct a given angle at M.

Step 3: Construct an angle 90° at N.

Step 4: Locate vertices L and O.

Step 5: Join L and O.

question 1. Construct a quadrilateral ADBC, in which AB = 4.4 cm, BC = 4 cm,  CD = 6.4 cm, DA = 2.8 cm, and BD = 6.6 cm. Measure the length of AC.

• With centre B and radius 4 cm, draw an arc. Then, with centre D and radius 6.4 cm, draw another arc which cuts the earlier arc of this step at the point C.
• Join BC and DC.

Thus, ABCD is the required quadrilateral.

Join AC. Length of AC = 6 cm. 

question 2. Construct a quadrilateral ABCD in which BC = 4 cm, CA = 5.6 cm, AD = 4.5 cm, CD = 5 cm, and BD = 6.5 cm.

• Join DA and CA.
• With centre D and radius 6.5 cm, draw an arc. Then with centre C, and radius 4 cm, draw another arc which cuts the earlier arc of this step at point B.
• Join CB and B.

Thus, ABCD is the required quadrilateral.

question 3. Construct the quadrilateral LIFT in which LI = 4 cm, IF = 3 cm, TL = 2.5 cm, LF = 4.5 cm and IT = 4 cm.

Solution:Steps of construction

• Take a line segment LF = 4.5 cm

question 4. Construct the quadrilateral GOLD in which OL = 7.5 cm, GL = 6 cm, GD = 6 cm, LD = 5 cm and OD = 10 cm.

• Now, with L and D as centre cut off two arcs of length 7.5 cm and 10 cm respectively to obtain a point O.
• Join OL, OD and OG.
•  Required quadrilateral GOLD is obtained.

CONSTRUCTING A QUADRILATERAL WHEN LENGTHS OF ITS ADJACENT SIDES AND THREE ANGLES ARE GIVEN:

Step 1: Draw a line segment of EF (say) of given length.

Step 2: Construct a given angle at E.

Step 3: Construct a given angle at F.

Step 4: Locate point G.

Step 5: Locate point H.

question 1. Construct the quadrilateral MORE in which MO = 6 cm, OR = 4.5 cm, ∠M = 60º, ∠O = 105º, ∠R = 105º.

Solution:

question 2. Construct a rectangle OKAY in which OK = 7 cm and KA = 5 cm.

Solution:

question 3. Construct a quadrilateral ABCD in which, AB = 4.3 cm, BC = 5 cm, ∠A = 60°, ∠B = 100° and ∠C = 125°.

Then, ABCD is the required quadrilateral.

question 4. (i) Can a quadrilateral PLAN be constructed if PL = 6 cm, LA = 9.5 cm,      ∠P = 75º, ∠L = 150º and ∠A = 140º.

(ii) In a parallelogram, the lengths of adjacent sides are known. Do we still need measures of the angles to construct the parallelogram?

Solution: (i) ∠P + ∠L + ∠A = 365º

75º + 150º + 140º = 365º

Sum of four angles in a quadrilateral is 360º.

∴ Quadrilateral is not possible to construct.

(ii) No, it is not required to have the measure of angles because opposite sides of a parallelogram are equal.

If AB and BC are given 

then, AB = DC, BC = AD.         

CONSTRUCTING A QUADRILATERAL WHEN LENGTHS OF ITS THREE SIDES AND TWO DIAGONALS ARE GIVEN:

STEPS:

Step1: Draw a rough sketch of the quadrilateral PQRS.

question 1. Construct a quadrilateral PQRS in which PQ = 3.5 cm, QR = 2.5 cm, RS = 4.1 cm, ∠Q = 75° and ∠R = 120°. Measure side PS.

Solution:Steps of construction

•  Draw a rough sketch of the quadrilateral PQRS.

• Join PS.

Thus, PQRS is the required quadrilateral.

Measuring PS we find that PS = 3.7 cm.

question 2. Construct a quadrilateral ABCD in which, AB = 4.2 cm, BC = 5 cm, CD = 5.3 cm, ∠B = 120° and ∠C = 75°.

Solution: Steps of construction Draw BC = 5 cm. Make ∠CBX = 120°. With B as centre and radius 4.2 cm draw an arc BA = 4.2 cm, so as it cuts BX at A. Make ∠BCY = 75°. With C as centre and radius 5.3 cm draw an arc so as it cuts CY at D.

•  Join DA.

Thus, ABCD is the required quadrilateral.

question 3.  Construct a quadrilateral ABCD given AB = 5.1 cm, AD = 4 cm, BC = 2.5 cm, ∠A = 60° and ∠B = 85°.

Sol. Steps of construction

• Join CD.

The quadrilateral ABCD so obtained is the required quadrilateral.

question 4. (i) Can a quadrilateral ABCD be constructed with AB = 5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and ∠B = 80º?

(ii) Can a quadrilateral PQRS be constructed with PQ = 4.5 cm, ∠P = 70º,   ∠Q = 100º, ∠R = 80º and ∠S = 110º?

Solution:(i)

We can construct a quadrilateral ABCD with four sides and one angle.

(ii)

Here, at least one side RQ or PS is required. So, data is in-sufficient. Construction is possible only if 

(a) two angles and two sides, e.g. if ∠A = 90º, ∠B = 90º, AD = 6 cm, BC = 5 cm. are given.

(b) four angles and two sides e.g. PQ = 4.5 cm, ∠P = 65º, ∠Q = 110º,              ∠QR = 4.7 cm ∠R = 115º are given.

SPECIAL CASES QUADRILTERALS

A quadrilateral can be constructed using properties of specified quadrilateral.

• Opposite sides are equal.
• Diagonals are equal.
• Each angle 90º.

RHOMBUS

• Diagonals bisect each other at 90º.
• All sides are equal.
• Diagonals are unequal.

STEPS:

Step 1: Draw a side of given leng

Step 3: Draw side CE (say) of length th (say) CL

Step 2: Draw side LU (say) of given length perpendicular to CL at L. equal to LU and perpendicular to CL at C.

Step 4: Draw side UE.

STEPS:

Step 1: Draw diagonal (say) AY and its perpendicular bisector.

Step 2:  Draw sides say AL and AZ of given length.

Step 3: Draw sides LY and YZ.