SPECIAL CASES QUADRILTERALS
Practical geometry of Class 8
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.
Solution: Steps of construction


 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.
Solution: Steps of construction


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


 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.
Solution: Steps of construction


 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.
Solution:Steps of construction


 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.
Solution: Here, GL = 6 cm GD = 6 cm and LD = 5 cm. Steps of construction


 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:
Steps of construction


question 2. Construct a rectangle OKAY in which OK = 7 cm and KA = 5 cm.
Solution:
Steps of construction


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


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.
Step 2: Draw a line segment of given length. Step 3: Construct the two given angles at two ends of the line. Step 4: With centre O and radius of the given length, draw an arc. Step 5: With centre R and radius of thje second side, draw another arc. Step 6: Join the two arcs 

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


 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 insufficient. 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.
PARALLELOGRAM
SQUARE
RECTANGLE 

 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.