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We have already learnt to construct parallel and perpendicular lines given a set of conditions. di, * Aline parallel to a given line’ from a point outside it © “A line parallel toa given line ata given sstanice, , , , , , We have also learned the construction of triangles given a minimum of three elements., , + When three Sides are given (SSS criterion) - 34/2; When two sides and the included angle‘are given, Zz (SAS criterion), Sem, 4cm t ic, 6cm, x Tem Y, , A 8cm— B, , * When two angles and the included side are given * When the hypotenuse and ¢ one side are given, , , , , , (ASA criterion) _(RHS criterion), y i, R, iy, 10cm, Exercise . fone 3, , , , 1, Construct a line parallel to the 4. Construct APOR with 7P = = 45°,, , , , SAR ‘ QP =7 cm, and, given line AB from a point 0 pa 20 =85°. :, outside it... 5. Construct ADEF with <D=75°, ZE = 53°, and, , 2. Construct.a line parallel to the given line / which is DE=6cm.:., , . ata distance of 5 cm from it. 6. Construct AABC with ee BC=7cm and, , 3, Construct AABC with AB = BC = AC = 6.5 cm. Z2B= 110°,

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coNSTRUCTING QUADRILATERALS, , metrical constructions that we have learnt §, ag will now be applied further to construction ae, , quadeilaterals., , , , , , jnimum Elements Required S, A quadrilateral has a total of ten elements whi h, include four sides, four angles, and two discon, To construct a unique quadrilateral we need the :, measurements of a minimum of five elements out, of ten. Note that we will need to know a specific, combination of five elements., Let us discuss the construction of quadrilaterals in, the following different cases., . Given four sides and a diagonal, ~ « Given four sides and an angle, + Given three sides and two diagonals, + Given three sides and two included angles, + Given two adjacent sides and three angles, We will always start by drawing a rough sketch of, the quadrilateral using the given dimensions., , , , Given Four Sides and a Diagonal, , --Brample I: Construct a quadrilateral ABCD in which, , , , , , AB=5cm, BC = 4m, CD = 3.5 em, AD = 4.2m,, and diagonal AC = 6cm, p 2S EC, Solution:, , dom, , raw a rough sketch, 420m, , Step 1: Di, given, , of quadrilateral ABCD,, ABaScm, BC=4em,, CD=3.5 cm, AD = 4,2 cm,, , and AC = 6 cm., , Step 2: Draw & line seg, , AB of length 5 ¢m, Step 3: With A as centre and radius 6m, draw an are, onone side of ABStep 4: With B as centre and, intersecting the previous arc arc., , , , A Sem 8, Rough sketch, , , , ent, , radius 4 cm draw an are, , Step 5: Join ‘AC and BC, : :, , Step 6: Now, with Cas centre and radius 3.5m, daw, an arc. :, , -Step 7: Draw another py ase C, , , , , , , are with A as the centre, and radius 4.2 ¢m, , ~ cutting the previous are, Atpoint D.Join AD., ‘Hence, ABCD is the ee, ~” required quadrilateral., , 420m jem, , Example 2: Construct a quadrilateral ABCD in which, AB =4.5 cm, BC =5 cm, CD =3cm,AD=5.5 cm,, and AC = 6 cm. eat ce, Solution: °° . De 3cm"', Step 1: Draw a rough sketch ee 4 oF f, of quadrilateral ABCD, given 38 on os i, , , < / Sem, , AB= aS, B=4.5 cm,BC=5cm, yi oe te, , CD=3cm,, AD =5.5,.cm, and AC =6 cm. Rough sketch, , Observe that the quadrilateral ABCD can beles AABC-and, , conveniently divided into two triangle, , AADC, with AC as the common arm., , , , , , nt AC of length 6cm., , Step 2: Draw a line segme, d radius 5.5 cm, draw an, , Step 3: With A as centre an, , arc on one side of AC., Step 4: With Cas centre and radius, , intersecting the previous are at D., Step 5: Join AD and CD., , Step 6: Now, with, Aas centre and, radius 4.5 cm,, draw an arc on the, other side of AC., Step 7: Draw, another ure with, Cas the centre, and radius Scm, interseeting the, previous arc at, point &,, Step 8: Join AB and CB., Hence, ABCD is the requii, , 3 cm draw an are, , , , , , red quadrilateral., , wipanatceneenn,, , , , I. Is it possible to construct & quadrilateral, PORS in which PQ=3em. QR= 4m,, RS = 5.2cm, SP = 6m, and QS =9cm? if, not, give reason., 2. Construct a quadrilateral LMNO in which, LM =4com,MN 26cm, NO = 5.2m,, OL=4cm, and LN = 8 cm., 3: Construct a quadrilateral ABCD in which, AB = 3.5m, BC =4cm, CD = DA =4,5 em., and diagonal BD = 5.4 cm. fee, , , , , , , , , Senne ee, , , , Sst Sayles aseadirnennecnnres metre

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Solution: ', , Step 1: Draw a rough, sketch of quadrilateral, BOAT, given BO = 4.cm,, OA = 3.6cm, AT = 4.2 cm,, BT =4.8 cm, and ZO=70°., Step 2: Draw a line, segment BO of length 4 cm., Step 3: Construct an angle of 70° at O., , Step 4: With O as centre and radius 3.6 cm draw an, arc intersecting OX at point A., , Step 5: With A as centre and radius 4.2cm draw an, are., , Step 6: Draw another., arc with B as the centre, and radius 4.8 cm, intersecting the arc, drawn previously at 7., Step 7: Join BT and AT., Hence, BOAT is the, required quadrilateral., , B 4com 0, , , , 1. Construct a quadrilateral ABCD in which, AB = BC =3.5 cm, AD= CD =5 cm, and, ZABC = 120°., , 2. Construct a quadrilateral ABCD in which, , AB =3cm, BC=3.4cm, CD =2.7 cm,, , DA =3.5 cm, and ZA = 70°., , Given Three Sides and Two Diagonals, , » Example 4: Construct ‘a quadrilateral MNOP in 1 which, “MN = 4 cm, NO =3.8 cm, MP =3.cm, diagonal, , MO =5cm, and diagonal eS 4. Oe cm., , Solution:, , _ Step 1: Draw a rough, sketch of quadrilateral., - MNOP, given MN = 4 cms:, NO =3.8 cm, MP =3 cm,, diagonal MO = 5 cm, and, diagonal NP = 4.6 cm., , Step 2: Draw a line, segment MN of length 4. cm., Step 3: With “Mas centre and radius 5 cm draw an arc, , above. the dae segment MN., , @ ert SESE, , , , M 4cm .. N, Rough sketch, , Rough sketch |=, , , , ‘Step 5: Draw, , , , , , , , , , , , , , , , , , , , , , , , , , , With N as centre and radius 3.8 cm draws, er arc intersecting the previous arc atO,, Step 5: Join ON., , Step 6; With M as centre and radius, arc., , Step 7: With N as centre, another arc intersecting the previo, Step 8: Join OP, and MP., , Hence, MNOP, is the required, quadrilateral., [Note:, Diagonals are, shown as dotted, lines], , 3 cm, draw an->, , and radius 4.6 cm, draw, usly drawn are at P,, a., , 1. Construct a quadrilateral ABCD in which, AB =3.6 cm, BC = 3.2 cm, AD= 2.9 cm,, diagonal AC = 4.5 cm, and diagonal, BD =4.2 cm., , 2. Construct a quadrilateral LMNO in which, LN=LO=6.5 cm, MN= 7cm,MO=9cm,, and NO =5 cm. Also, find the length of the, remaining side., , , , Given Three Sides and Two Included, Angles, , Example 5: Construct a quadrilateral NEST in which., NE =3.8 cm, ZNES = 75°, ES = 4.2 cm,, , ZENT = 110°, and NT = 4.4 cm., , Solution: T., , Step 1: Draw a rough, , sketch of the quadrilateral, NEST, given NE = 3.8 cm,., ZNES = 75°, ES = 4.2 cm, —, ZENT = 110°, and 2, NT=4.4 cm. N, Step 2: Draw a line, , segment NE of length 3.8 cm., Step 3: Construct Y : ee ¢, ZNES = 75° at E. eee TEN, Step 4: With E as, centre and radius, 4.2 cm, draw an are, intersecting EX at S., , , , , , , , , , , , , , , 44cm, , 38em. =, Rough sketch, , ZENT = 110° at N.

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. With V as centre ., sep : and radius 4., SM persecting NY at T and join 7s, 4 cm, draw an, the quadrilateral NEST thus obtained, is th, , is the, , eo pence :, : quired quadrilateral. t, , , , , , , 1, Construct a quadrilateral MORE i ay, MO =3.5 cm, OR = 4.8 cm, RE ee i, ZO = 125°, and ZR = 60°. cm, i, 2. Construct a quadrilateral SENT in whi, sE=70m, EN=5.8 cm, NT= 2.5 en, ZS = 45°, and ZE = 90°. ,, , , , , , , Given two. adjacent sides and three, , angles, Fyample 6: Construct a quadrilateral SALE in which, go45m, ZSAL = 100°, AL=4.cm,, , ZALE = 90°, and ZASE = 60°., ‘Solution:, , Step 1: Draw a rough sketch, of the quadrilateral SALE,, , given SA = 4.5m,, , ZSAL = 100°, AL = 4 cm,, ZALE = 90°, and ZASE = 60°., Step 2: Draw a line segment, SA of length 4.5 cm., , Step 3: Construct an ang!, 100° at A., , Step 4: With A as centre, and radius 4 cm, draw, anare cutting AX at L., Step 5: Construct an, angle of 90° at L., , Step 6; Construct the, angle ZASZ = 60° at, 5.Name the point of, intersection of LY and, WasEs take ee, Hence, SALE is the required quadrilateral:, , , , S 45cm, Rough sketch, , le of, , , , 4.5 cm:, , , , , , , 1; Construct a quadril, PL 260m, LA=45, YL = 105°, and: ZA, , 2.18 it possible, ABCD in which BC=, , ty 295%, ZC = 120% and 2D, Teasons. me, , , , , , , , , , , CONSTRUCTION OF A SQUARE, , We have studied that to construct a unique, , quadrilateral, a minimum of five specific elements, , are required to be known. However, we will now, , sce that by using their properties, it is possible, , to Construct certain quadrilaterals given less than, , five element measures., , Square is a quadrilateral in which all the sides are, , of equal length and each angle is a right angle., , re 6 recall the basic properties of a square., , « All the sides of a square are at, equal. : y i Zl., , I, « All the angles of a square 2 om L, , , , are equal to 90°. | i, © The diagonals are equaland 3 we “gic, perpendicular bisectors of, one another. ‘, Using the properties of a square, let us try to, construct it given some specific parameters., When One Side is Given, Example 7: Construct a square of side 5 cm., Solution: ?, Step 1: Let us draw a rough A 5om_D., sketch of the square ABCD, with the side length 5 cm. 5cem 5cm, We have, AB = BC = cD=, DA=5 cmand angles ZA =, ZB = ZC = ZD=90°. B sem C, Step 2: Draw a line segment Rough sketch, AB of length 5 cm., Step 3: With the help of compass, construct a, perpendicular at point A by taking Aas the centre and, dius less than half of AB., , a convenient ra, Step 4: Similarly,, , a perpendicular at point B., Step 5: With A and B as centres and radius 5 cm, draw, , arcs which intersect the perpendiculars at D and C, respectively. me ARES, Step 6: Join C and D., Hence, ABCD is the, required square with, side 5 cm., , (Note: Use a ruler, and check whether, , CD is Sem or not,, , If itis 5.cm, the, construction is, correct), , with the help of compass, construct, , , , , , , , os, 22. |Sem

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eo SL SUR IVES Ore’ Given =, Sort MeL OPRE er i ‘, HoH: Wy seuRsttty ee OEE Examp 9: Construct a rectangle ABCD wi ea, iagonaly fs Mn at, the D Sons 5M and 4 em, ae, Of a square are ; Solution: : ;, and icular nee equal, sae gm, oad Perpendicular bisector of oR Step 1: Draw a rough D, © Oe another. To Construct the A ne Sh Sketch of the Tectangle, Fequired Square ‘Wwe will follow: | 8em > f ABCD in which adjacent, BS below given siege sae Fe aes Measure 5 cm and, 8) ~ Draw the roy hes B cm,, x Sket i tir 25 zB, the requi aaa a cn ch of | Rough sketeh We can write this as,AB 4, Step 2: D, li . =CD=5om and Rough sketeh,, Step 3: win ne segment AC of length g cm, BC=DA=4cm, :, perpe di 1 ie help f compass, Construct a Step 2: Draw a line segment AB of length 5 cm,, ae pene bisector Of the line Segment AC. Mark Step 3: With the help of a compass, construct a, ' Intersection fi Perpendicular AX at A., i Step 2 Wii aS the centre and OA = 4 om as the Step 4: Similarly, with the help Of a compass,, Tadius, Cut the Perpendicular bisectors on either side construct a perpendicular BY at B. 3, Of the line Segment iD Step 5: With A and B as Centres and radius 4 om, day, AC at Band p BY arcs which intersect the Perpendiculdrs AX ang Bra, Tespectively, . Dandc respectively, 3 q, ‘Step 5: Join AB, BC, ft x Step 6: Join C and D,, , , and DA. Gee ae Be si = Hence, ABCD, Hence, ABCD ig the ., , \ 5 is the required, Tequired square. y iG: Tectangle with, Note: Measure the | A adjacent sides, lengths of AB, BC, a 7 Sm and 4 cm,, CD, and DA. All of ery (ote: Use the, these should be of SU heg a et ruler and check, “same length.), , Whether CD is, , Sem or not. if, , it is Som, the, , poreer a 28 od construction is., > correct), , 1. Construct a Square PORS with PQ=4cm., , 2 Construct a Square DART with DR = 10cm., , , , , , , Given One Side and a Diagonal, Example 10; Cc, , , , , , , , , , ; Onstruct a rectangle PQRS in which, side OR =5 cm and diagonal QS = 6.2 om., CONSTRUCTION OF A RECTANGLE Solio,, The properties of a rectangle are: Step 1: Draw a rough sketch of us, * Opposite sides of a rectangle are equal. rectangle PORS with QR=Scm |, * Each angle in a. E. and QS = 6.2 cm: : oo, rectangle is a right ae ae Draw a line Segment QR of S oh, : length 5 cm. oe, angle. : : :, ° Dugenals are equal in eee Construct ‘a Perpendicular sat ’, : : Rough sketch, eee ip of these Step 4: With Q as centre and radius " 4, With the help o:, , fare 6.2cm draw an arc, cutting RX at. on, properties, let us construct a rectangle given only Step 5: Draw an ‘are with ‘S'as the centre and ae ., : “specific dimensions... Ge Sem : ;