Chapter 4 Practical Geometry

Given:

A quadrilateral PQRS with the following side lengths and diagonal length:

PQ = 4 cm

QR = 6 cm

RS = 5 cm

PS = 5.5 cm

PR = 7 cm

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To Construct:

The quadrilateral PQRS.

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Construction Steps:

We are given the lengths of four sides and one diagonal. The diagonal PR divides the quadrilateral into two triangles, $\triangle$PQR and $\triangle$PSR. We will first construct $\triangle$PQR and then $\triangle$PSR.

Step 1: Draw a line segment PR of length 7 cm. This will be the diagonal of the quadrilateral.

Step 2: To locate vertex Q, take P as the center and draw an arc with a radius of 4 cm (length of PQ). Then, take R as the center and draw another arc with a radius of 6 cm (length of QR). The point where these two arcs intersect is the vertex Q.

Step 3: Join points P and Q, and points R and Q to form the sides PQ and QR. This completes the triangle $\triangle$PQR.

Step 4: To locate vertex S, take P as the center again and draw an arc with a radius of 5.5 cm (length of PS) on the other side of the diagonal PR. Then, take R as the center and draw an arc with a radius of 5 cm (length of RS). The point where these two arcs intersect is the vertex S.

Step 5: Join points P and S, and points R and S to form the sides PS and RS. This completes the triangle $\triangle$PSR.

The figure PQRS is the required quadrilateral.

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Final Construction Diagram:

(i) Construction of Quadrilateral ABCD:

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

To Construct: Quadrilateral ABCD.

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Construction Steps:

Step 1: Draw a line segment AC of length 7 cm, which will serve as the diagonal.

Step 2: With A as the center, draw an arc with a radius of 4.5 cm (equal to AB).

Step 3: With C as the center, draw an arc with a radius of 5.5 cm (equal to BC) to intersect the previous arc. The intersection point is B.

Step 4: Join AB and BC. This completes $\triangle$ABC.

Step 5: With A as the center, draw an arc with a radius of 6 cm (equal to AD) on the other side of AC.

Step 6: With C as the center, draw an arc with a radius of 4 cm (equal to CD) to intersect the arc from Step 5. The intersection point is D.

Step 7: Join AD and CD. This completes $\triangle$ADC.

ABCD is the required quadrilateral.

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(ii) Construction of Quadrilateral JUMP:

Given: JU = 3.5 cm, UM = 4 cm, MP = 5 cm, PJ = 4.5 cm, PU = 6.5 cm.

To Construct: Quadrilateral JUMP.

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Construction Steps:

Step 1: Draw a line segment PU of length 6.5 cm, which will serve as the diagonal.

Step 2: With P as the center, draw an arc with a radius of 4.5 cm (equal to PJ).

Step 3: With U as the center, draw an arc with a radius of 3.5 cm (equal to JU) to intersect the previous arc. The intersection point is J.

Step 4: Join PJ and JU. This completes $\triangle$PJU.

Step 5: With P as the center, draw an arc with a radius of 5 cm (equal to MP) on the other side of PU.

Step 6: With U as the center, draw an arc with a radius of 4 cm (equal to UM) to intersect the arc from Step 5. The intersection point is M.

Step 7: Join PM and UM. This completes $\triangle$PMU.

JUMP is the required quadrilateral.

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(iii) Construction of Parallelogram MORE:

Given: OR = 6 cm, RE = 4.5 cm, EO = 7.5 cm.

Properties of Parallelogram: Opposite sides are equal (MO = RE and ME = OR).

So, we derive that MO = 4.5 cm and ME = 6 cm.

To Construct: Parallelogram MORE.

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Construction Steps:

Step 1: Draw a line segment OR of length 6 cm.

Step 2: With O as the center, draw an arc with a radius of 7.5 cm (diagonal EO).

Step 3: With R as the center, draw an arc with a radius of 4.5 cm (side RE) to intersect the previous arc. The intersection point is E.

Step 4: Join RE and OE. This forms $\triangle$ORE.

Step 5: With E as the center, draw an arc with a radius of 6 cm (side ME, opposite to OR).

Step 6: With O as the center, draw an arc with a radius of 4.5 cm (side MO, opposite to RE) to intersect the previous arc. The intersection point is M.

Step 7: Join OM and ME.

MORE is the required parallelogram.

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(iv) Construction of Rhombus BEST:

Given: BE = 4.5 cm, ET = 6 cm.

Properties of Rhombus: All sides are equal (BE = ES = ST = TB).

Since BE = 4.5 cm, all sides of the rhombus are 4.5 cm. ET = 6 cm is a diagonal.

To Construct: Rhombus BEST.

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Construction Steps:

Step 1: Draw a line segment ET of length 6 cm (the diagonal).

Step 2: With E as the center and a radius of 4.5 cm, draw an arc on one side of ET.

Step 3: With T as the center and the same radius of 4.5 cm, draw another arc to intersect the first arc. The intersection point is B.

Step 4: Join EB and TB.

Step 5: Repeat steps 2 and 3 on the other side of ET to locate the point S.

Step 6: Join ES and TS.

BEST is the required rhombus.

Given:

A quadrilateral ABCD with the following side lengths and diagonal lengths:

BC = 4.5 cm

AD = 5.5 cm

CD = 5 cm

Diagonal AC = 5.5 cm

Diagonal BD = 7 cm

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To Construct:

The quadrilateral ABCD.

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Construction Steps:

We are given the lengths of three sides (BC, CD, AD) and two diagonals (AC, BD). We can begin by constructing one of the triangles for which we have three known lengths, for example, $\triangle$ADC or $\triangle$BCD.

Step 1: Draw a line segment CD of length 5 cm.

Step 2: To locate vertex A, we use the lengths AC = 5.5 cm and AD = 5.5 cm. With C as the center, draw an arc with a radius of 5.5 cm. With D as the center, draw another arc with a radius of 5.5 cm. The intersection point of these two arcs is A.

Step 3: Join AC and AD. This completes $\triangle$ADC.

Step 4: To locate vertex B, we use the lengths BC = 4.5 cm and BD = 7 cm. With C as the center, draw an arc with a radius of 4.5 cm on the other side of the triangle. With D as the center, draw an arc with a radius of 7 cm to intersect the previous arc. The intersection point is B.

Step 5: Join BC and BD. Also, join the final side AB.

ABCD is the required quadrilateral.

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Final Construction Diagram:

(i) Construction of Quadrilateral LIFT:

Given: LI = 4 cm, IF = 3 cm, TL = 2.5 cm, LF = 4.5 cm, IT = 4 cm.

We are given the lengths of three sides (LI, IF, TL) and two diagonals (LF, IT).

To Construct: Quadrilateral LIFT.

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Construction Steps:

Step 1: Draw a line segment LI of length 4 cm.

Step 2: With L as the center, draw an arc with a radius of 2.5 cm (length of TL).

Step 3: With I as the center, draw an arc with a radius of 4 cm (length of diagonal IT) to intersect the previous arc. The intersection point is T.

Step 4: Join LT and IT.

Step 5: With I as the center, draw an arc with a radius of 3 cm (length of IF).

Step 6: With L as the center, draw an arc with a radius of 4.5 cm (length of diagonal LF) to intersect the previous arc. The intersection point is F.

Step 7: Join IF, LF, and FT.

LIFT is the required quadrilateral.

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(ii) Construction of Quadrilateral GOLD:

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

We are given the lengths of three sides (OL, LD, GD) and two diagonals (GL, OD).

To Construct: Quadrilateral GOLD.

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Construction Steps:

Step 1: Draw a line segment LD of length 5 cm.

Step 2: With L as the center, draw an arc with a radius of 6 cm (diagonal GL).

Step 3: With D as the center, draw an arc with a radius of 6 cm (side GD) to intersect the previous arc. The intersection point is G.

Step 4: Join LG and DG.

Step 5: With D as the center, draw an arc with a radius of 10 cm (diagonal OD).

Step 6: With L as the center, draw an arc with a radius of 7.5 cm (side OL) to intersect the previous arc. The intersection point is O.

Step 7: Join DO, OL, and OG.

GOLD is the required quadrilateral.

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(iii) Construction of Rhombus BEND:

Given: Diagonal BN = 5.6 cm, Diagonal DE = 6.5 cm.

Properties of Rhombus: Diagonals bisect each other at right angles ($90^\circ$).

To Construct: Rhombus BEND.

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Construction Steps:

Step 1: Draw a line segment DE of length 6.5 cm (one diagonal).

Step 2: Construct the perpendicular bisector of DE. Let the point where the bisector intersects DE be O.

Step 3: The other diagonal BN lies on this perpendicular bisector. We need to mark points B and N on it. The total length of BN is 5.6 cm. Since O is the midpoint, the length from O to B is half of this, and from O to N is the other half.

Length OB = ON = $\frac{1}{2} \times 5.6$ cm = 2.8 cm.

Step 4: With O as the center, draw an arc with a radius of 2.8 cm to cut the perpendicular bisector above DE. Mark this point as B.

Step 5: With O as the center, draw another arc with a radius of 2.8 cm to cut the perpendicular bisector below DE. Mark this point as N.

Step 6: Join the points D, B, E, and N to form the rhombus.

BEND is the required rhombus.

Given:

A quadrilateral MIST with the following measurements:

MI = $3.5$ cm

IS = $6.5$ cm

$\angle$M = $75^\circ$

$\angle$I = $105^\circ$

$\angle$S = $120^\circ$

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To Construct:

The quadrilateral MIST.

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Construction Steps:

We are given two adjacent sides and three angles. We can start the construction from one of the given sides.

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

Step 2: At point M, use a protractor to construct an angle of $75^\circ$. Draw a ray MX from M.

Step 3: At point I, use a protractor to construct an angle of $105^\circ$. Draw a ray IY from I.

Step 4: With I as the center and a radius of $6.5$ cm, draw an arc that cuts the ray IY. Mark the intersection point as S.

Step 5: At point S, use a protractor to construct an angle of $120^\circ$. Draw a ray SZ from S. This ray should intersect the ray MX from Step 2.

Step 6: The point where the ray SZ intersects the ray MX is the vertex T.

MIST is the required quadrilateral.

(Verification: The sum of angles in a quadrilateral is $360^\circ$. So, $\angle M + \angle I + \angle S + \angle T = 75^\circ + 105^\circ + 120^\circ + \angle T = 300^\circ + \angle T = 360^\circ$. This gives $\angle T = 60^\circ$. You can measure the angle at T in your constructed figure to check if it is $60^\circ$.)

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Final Construction Diagram:

(i) Construction of Quadrilateral MORE:

Given: MO = 6 cm, OR = 4.5 cm, $\angle$M = $60^\circ$, $\angle$O = $105^\circ$, $\angle$R = $105^\circ$.

To Construct: Quadrilateral MORE.

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Construction Steps:

Step 1: Draw a line segment MO of length 6 cm.

Step 2: At point M, use a protractor to construct an angle of $60^\circ$. Draw a ray from M.

Step 3: At point O, use a protractor to construct an angle of $105^\circ$. Draw a ray from O.

Step 4: With O as the center and a radius of 4.5 cm, draw an arc that cuts the ray from O. Mark this point as R.

Step 5: At point R, use a protractor to construct an angle of $105^\circ$. Draw a ray from R such that it intersects the ray drawn from M.

Step 6: The intersection point of the rays from M and R is the vertex E.

MORE is the required quadrilateral.

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(ii) Construction of Quadrilateral PLAN:

Given: PL = 4 cm, LA = 6.5 cm, $\angle$P = $90^\circ$, $\angle$A = $110^\circ$, $\angle$N = $85^\circ$.

First, we find the fourth angle, $\angle$L, using the angle sum property of a quadrilateral.

$\angle$L = $360^\circ - (\angle P + \angle A + \angle N)$

$\angle$L = $360^\circ - (90^\circ + 110^\circ + 85^\circ)$

$\angle$L = $360^\circ - 285^\circ = 75^\circ$

To Construct: Quadrilateral PLAN.

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Construction Steps:

Step 1: Draw a line segment PL of length 4 cm.

Step 2: At point P, use a protractor to construct a right angle ($90^\circ$). Draw a ray from P.

Step 3: At point L, use a protractor to construct an angle of $75^\circ$. Draw a ray from L.

Step 4: With L as the center and a radius of 6.5 cm, draw an arc that cuts the ray from L. Mark this point as A.

Step 5: At point A, use a protractor to construct an angle of $110^\circ$. Draw a ray from A such that it intersects the ray drawn from P.

Step 6: The intersection point of the rays from P and A is the vertex N.

PLAN is the required quadrilateral.

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(iii) Construction of Parallelogram HEAR:

Given: HE = 5 cm, EA = 6 cm, $\angle$R = $85^\circ$.

Properties of Parallelogram: Opposite angles are equal, so $\angle$E = $\angle$R = $85^\circ$. Adjacent angles are supplementary, so $\angle$H = $180^\circ - 85^\circ = 95^\circ$.

To Construct: Parallelogram HEAR.

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Construction Steps:

Step 1: Draw a line segment HE of length 5 cm.

Step 2: At point E, construct an angle of $85^\circ$. Draw a ray from E.

Step 3: With E as the center and a radius of 6 cm, draw an arc that cuts the ray from E. Mark this point as A.

Step 4: With A as the center, draw an arc with a radius of 5 cm (since AR = HE = 5 cm).

Step 5: With H as the center, draw an arc with a radius of 6 cm (since HR = EA = 6 cm) to intersect the previous arc. Mark this point as R.

Step 6: Join AR and HR.

HEAR is the required parallelogram.

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(iv) Construction of Rectangle OKAY:

Given: OK = 7 cm, KA = 5 cm.

Properties of Rectangle: All interior angles are $90^\circ$. Opposite sides are equal.

To Construct: Rectangle OKAY.

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Construction Steps:

Step 1: Draw a line segment OK of length 7 cm.

Step 2: At point O, construct a right angle ($90^\circ$) and draw a ray.

Step 3: At point K, construct a right angle ($90^\circ$) and draw a ray.

Step 4: With K as the center and a radius of 5 cm, draw an arc that cuts the ray from K. Mark this point as A.

Step 5: With O as the center and a radius of 5 cm, draw an arc that cuts the ray from O. Mark this point as Y.

Step 6: Join AY.

OKAY is the required rectangle.

Given:

A quadrilateral ABCD with the following measurements:

AB = 4 cm

BC = 5 cm

CD = 6.5 cm

$\angle$B = $105^\circ$

$\angle$C = $80^\circ$

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To Construct:

The quadrilateral ABCD.

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Construction Steps:

We are given three sides and two included angles. We will start by drawing the side BC, which is between the two given angles.

Step 1: Draw a line segment BC of length 5 cm.

Step 2: At point B, use a protractor to construct an angle of $105^\circ$. Draw a ray BX.

Step 3: With B as the center and a radius of 4 cm, draw an arc that cuts the ray BX. Mark this point as A.

Step 4: At point C, use a protractor to construct an angle of $80^\circ$. Draw a ray CY.

Step 5: With C as the center and a radius of 6.5 cm, draw an arc that cuts the ray CY. Mark this point as D.

Step 6: Join points A and D.

ABCD is the required quadrilateral.

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Final Construction Diagram:

(i) Construction of Quadrilateral DEAR:

Given: DE = 4 cm, EA = 5 cm, AR = 4.5 cm, $\angle$E = $60^\circ$, $\angle$A = $90^\circ$.

To Construct: Quadrilateral DEAR.

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Construction Steps:

We are given two adjacent sides (DE, EA), the included angle ($\angle$E), another adjacent side (AR), and an angle ($\angle$A) adjacent to the second side.

Step 1: Draw a line segment EA of length 5 cm.

Step 2: At point E, construct an angle of $60^\circ$ using a protractor. Draw a ray from E.

Step 3: With E as the center and a radius of 4 cm, draw an arc that cuts the ray from E. Mark this point as D.

Step 4: At point A, construct a right angle ($90^\circ$) using a protractor. Draw a ray from A.

Step 5: With A as the center and a radius of 4.5 cm, draw an arc that cuts the ray from A. Mark this point as R.

Step 6: Join points D and R.

DEAR is the required quadrilateral.

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(ii) Construction of Quadrilateral TRUE:

Given: TR = 3.5 cm, RU = 3 cm, UE = 4 cm, $\angle$R = $75^\circ$, $\angle$U = $120^\circ$.

To Construct: Quadrilateral TRUE.

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Construction Steps:

We are given three consecutive sides and the two angles included between them.

Step 1: Draw a line segment RU of length 3 cm.

Step 2: At point R, construct an angle of $75^\circ$ using a protractor. Draw a ray from R.

Step 3: With R as the center and a radius of 3.5 cm, draw an arc that cuts the ray from R. Mark this point as T.

Step 4: At point U, construct an angle of $120^\circ$ using a protractor. Draw a ray from U.

Step 5: With U as the center and a radius of 4 cm, draw an arc that cuts the ray from U. Mark this point as E.

Step 6: Join points T and E.

TRUE is the required quadrilateral.

Given:

Side length of the square = 4.5 cm.

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To Construct:

A square with side length 4.5 cm.

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Construction Steps:

A square is a quadrilateral with four sides of equal length and four right angles ($90^\circ$).

Step 1: Draw a line segment AB of length 4.5 cm.

Step 2: At point A, construct a right angle ($90^\circ$). Draw a ray AX perpendicular to AB.

Step 3: With A as the center and a radius of 4.5 cm, draw an arc that cuts the ray AX. Mark this point as D.

Step 4: With B as the center, draw an arc with a radius of 4.5 cm.

Step 5: With D as the center, draw another arc with a radius of 4.5 cm to intersect the previous arc. Mark this intersection point as C.

Step 6: Join BC and DC.

ABCD is the required square.

Given:

A rhombus ABCD.

Length of diagonal AC = 6 cm.

Length of diagonal BD = 7 cm.

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To Check Possibility:

Is it possible to construct a rhombus with the given diagonal lengths?

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Justification:

Yes, it is possible to construct the rhombus.

We can justify this by using the key properties of the diagonals of a rhombus:

1. The diagonals bisect each other.

2. The diagonals are perpendicular to each other (they intersect at $90^\circ$).

Let the diagonals AC and BD intersect at a point O. Because the diagonals are perpendicular bisectors of each other, they form four congruent right-angled triangles.

Let's consider one of these triangles, for example, $\triangle$AOB.

Since the diagonals bisect each other:

Since the diagonals are perpendicular, $\triangle$AOB is a right-angled triangle with the right angle at O. We can find the length of the side of the rhombus (which is the hypotenuse AB) using the Pythagorean theorem.

$AB^2 = (3)^2 + (3.5)^2$

$AB^2 = 9 + 12.25$

$AB^2 = 21.25$

$AB = \sqrt{21.25}$ cm

Since we can find a unique, positive value for the side length AB, a rhombus with these diagonal lengths can exist. The construction is possible because we can draw one diagonal, construct its perpendicular bisector, and then mark the endpoints of the second diagonal before connecting the vertices.

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Conclusion:

Yes, it is possible to construct a rhombus with diagonals of length 6 cm and 7 cm because these lengths define a unique, valid side length for the rhombus.

Given:

A square named READ with side length RE = 5.1 cm.

Properties: In a square, all sides are equal (RE = EA = AD = DR = 5.1 cm) and all interior angles are $90^\circ$.

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To Construct:

The square READ.

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Construction Steps:

Step 1: Draw a line segment RE of length 5.1 cm.

Step 2: At point R, construct a right angle ($90^\circ$). Draw a ray from R.

Step 3: At point E, construct a right angle ($90^\circ$). Draw a ray from E.

Step 4: With R as the center and a radius of 5.1 cm, draw an arc that cuts the ray from R. Mark this point as D.

Step 5: With E as the center and a radius of 5.1 cm, draw an arc that cuts the ray from E. Mark this point as A.

Step 6: Join D to A.

READ is the required square.

Given:

A rhombus with diagonal lengths $d_1 = 5.2$ cm and $d_2 = 6.4$ cm.

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To Construct:

A rhombus with the given diagonal lengths.

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Properties of a Rhombus: The diagonals bisect each other at right angles ($90^\circ$).

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Construction Steps:

Step 1: Draw a line segment AC of length 6.4 cm (one diagonal).

Step 2: Construct the perpendicular bisector of AC. Let the point where the bisector intersects AC be O.

Step 3: The second diagonal BD has a total length of 5.2 cm. Since it is bisected by O, the length from O to B is half of this, and from O to D is the other half. So, OB = OD = $\frac{1}{2} \times 5.2$ cm = 2.6 cm.

Step 4: With O as the center, draw an arc with a radius of 2.6 cm to cut the perpendicular bisector above AC. Mark this point as B.

Step 5: With O as the center, draw another arc with a radius of 2.6 cm to cut the perpendicular bisector below AC. Mark this point as D.

Step 6: Join the points A, B, C, and D.

ABCD is the required rhombus.

Given:

A rectangle with adjacent sides of lengths 5 cm and 4 cm.

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To Construct:

A rectangle with the given side lengths.

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Properties of a Rectangle: All interior angles are $90^\circ$.

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Construction Steps:

Step 1: Draw a line segment AB of length 5 cm.

Step 2: At point A, construct a right angle ($90^\circ$). Draw a ray from A.

Step 3: With A as the center and a radius of 4 cm, draw an arc that cuts the ray from A. Mark this point as D.

Step 4: With B as the center, draw an arc with a radius of 4 cm.

Step 5: With D as the center, draw an arc with a radius of 5 cm to intersect the previous arc. Mark this point as C.

Step 6: Join BC and DC.

ABCD is the required rectangle.

Given:

A parallelogram OKAY with adjacent sides OK = 5.5 cm and KA = 4.2 cm.

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Construction and Uniqueness:

To construct a parallelogram, we are given the lengths of two adjacent sides. However, this information is not enough to define a unique shape. The angle between the two sides OK and KA can be changed, which would result in different parallelograms.

For example, let's construct one possible parallelogram by choosing an angle.

Step 1: Draw a line segment OK of length 5.5 cm.

Step 2: At point K, choose an arbitrary angle (e.g., $60^\circ$) and draw a ray.

Step 3: With K as the center and a radius of 4.2 cm, draw an arc to cut the ray. Mark this point as A.

Step 4: With A as the center, draw an arc with a radius of 5.5 cm (since AY = OK).

Step 5: With O as the center, draw an arc with a radius of 4.2 cm (since OY = KA) to intersect the previous arc. Mark this point as Y.

Step 6: Join AY and OY.

This gives us one possible parallelogram.

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Is it unique?

No, the parallelogram is not unique.

If we had chosen a different angle at K in Step 2 (for example, $75^\circ$ or $90^\circ$), we would have constructed a different parallelogram. Since the angle between the adjacent sides is not specified, an infinite number of different parallelograms can be drawn with the given side lengths.

Therefore, to construct a unique parallelogram, we need one more piece of information, like an interior angle or the length of a diagonal.