Practical Geometry

(i) Firstly we draw a rough sketch of the sum.

Step 1: First we draw the line segment AC with measurement 7.5cm

Step 2: Taking A as the center, we draw an arc of 4.5cm and taking C as a center we draw another arc such that they are intersecting.

Join AB and AC.

Step 3: Again taking A and C as a center we draw an arc of 6 cm and 7 cm respectively.

Step 4: Join AD and DC.

We get the required quadrilateral ABCD.

(ii) Firstly we draw a rough sketch of the sum.

Step 1: First we draw the line segment PU with measurement 6.5cm

Step 2: Taking P as the center, we draw an arc of 5cm and taking U as a center we draw another arc of 6.5 cm such that they are intersecting.

Step 3: Join PM and MU.

Step 3: Again taking P and U as a center we draw an arc of 4 cm and 3.5 cm respectively intersecting at J.

Step 4: Join PJ and PU.

(iii) Firstly we draw a rough sketch of the sum.

Step 1: First draw the base 0E= 7.5 cm

Step 2: Taking O as a center, we draw an arc of 4.5 cm . Again taking E as a center we draw an arc of 6 cm such that they are intersecting at point M.

Step 3: Join OM and ME.

Step 4: Taking O as a center, we draw an arc of 6 cm. Again taking E as a center we draw an arc of 4.5 cm such that they are intersecting at point R.

Step 5: Join OR and OE.

(iv) We know that all sides of a rhombus are of the same length.

Hence, BE = ES = ST = TB.

A rough diagram for the sum.

Step 1: First, we will draw the base for the diagram.

Step 2: Taking 4.5 cm each from E and T we draw the arc such that intersecting each other at the point B.

Step 3: Joining BE and BT.

Step 4: Taking 4.5 cm each from E and T we draw the arc such that intersecting each other at the point T.

Step 5: Joining ES and TS.

(i) A rough sketch of the quadrilateral can be drawn to make the sum easier.

Step 1: First we will draw the base LT = 2.5 cm.

Step 2: Taking T and L as center we draw the arcs each measuring of 4 cm intersecting at point I. Join the points TI and IL.

Step 3: Now, taking L and I as center we draw the arcs measuring of 4.5 cm and 3 cm intersecting at point F. Join the points LF and FI.

(ii) Firstly we draw a rough sketch of the sum.

Step 1: Firstly we will draw the base for our diagram , OD = 10 cm.

Step 2: Taking G as a center we draw an arc of measuring 10 cm and take D as a center we draw an arc of length 5 cm such that both are intersecting at the point L.

Join the points GL and LD.

Step 3: Taking D as a center we draw an arc of measuring 6 cm and taking L as a center we draw an arc of length 7.5 cm such that both are intersecting at the point O.

Join the points DO and LO.

Step 4: Join OG.

(iii) Firstly, we draw a rough sketch of the sum.

Step 1: Draw a line segment BN of 5.6 cm.

Step 2: Draw a perpendicular bisector of BN.

[Since diagonal of a rhombus bisect perpendicularly. ]

Step 3: Taking O as the center, draw arcs measuring 3.25 cm such that they intersect the perpendicular bisector at point D and E.

Step 4: Joining the points BD, BE, EN and DN.

(i) The steps of construction are:

Step (a) Draw a line segment MO of 6 cm and an angle of 105° at point O.

As vertex R is 4.5 cm away from the vertex O, therefore, cut a line segment OR of 4.5 cm from this ray

Step (b) Now, draw an angle of 105° at point R

Step (c) Draw an angle of 60° at point M and this ray meet the previously drawn ray from R at point E

Thus, MORE is the required quadrilateral, as given below,

(ii) As we know that the sum of the angles of a quadrilateral is 360°

In quadrilateral PLAN,

∠ P + ∠ L + ∠ A + ∠ N = 360°

90° + ∠ L + 110° + 85° = 360°

285° + ∠ L = 360°

∠ L = 360° - 285°

= 75°

The steps of construction are:

Step (a) Draw a line segment PL of 4 cm and also draw an angle of 75° at point L. Vertex A is 6.5 cm away from vertex L so cut a line segment LA of 6.5 cm from this ray

Step (b) From point A draw an angle of 110°

Step (c) Draw an angle of 90° at point P. This ray will meet the previously drawn ray from A at point N

Thus, the required quadrilateral is shown below:

(iii) The steps of construction are:

Step (a) Draw a line segment HE of 5 cm and an angle of 85° at point E.

Vertex A is 6 cm away from vertex E, cut a line segment EA of 6 cm from this ray

Step (b) Vertex R is 6 cm and 5 cm away from vertex H and A respectively.

So by taking radius as 6 cm and 5 cm, draw arcs from point H and A respectively. These will intersect each other at point R

Step (c) Join R to H and A

Thu, the required quadrilateral is shown below:

(iv) Rectangle OKAY

OK = 7 cm

KA = 5 cm

Step (a) Draw a line segment OK of 7 cm and an angle of 90° at point K

Vertex A is 5 cm away from vertex K, cut a line segment KA of 5 cm from this ray

Step (b) Vertex Y is of 5 cm and 7 cm away from vertex O and A respectively. Now by taking 5 cm and 7 cm as radius draw arcs from point O and A respectively and these will intersect at point Y

Step (c) Join Y to A and O

Thus, the required quadrilateral is shown below:

(i) A rough sketch can be drawn.

Step 1: Draw DE of 5 cm.

[We will take AE as base since both points have angles so that it will get easier in the later part.]

Step 2: At point E and A we draw 60° and 90° respectively.

Step 3: From point D and E, we draw arcs of measurement of 4cm and 4.5 cm respectively such that they intersect at point R and A.

Step 4: Join AR.

(ii) A rough diagram will make it easier to do construction.

Step 1:

We will draw base of RU = 3 cm

Step 2: At point R and U we will draw angles of 75° and 120° respectively.

Step 3: Taking point R and U as center we draw arcs measuring of 3.5 cm and 4 cm respectively.

Step 4: Join TE.

All the sides of a square are the same.

Hence, RE=EA=AD=DR=5.1

Step 1: Draw RE= 5.1 cm

Step 2: Draw 90 at both points R and E.

[Since we know that all the sides of the square are perpendicular to each other]

Step 3: Taking R and E as a center we draw arcs of 5.1 cm such that they intersect at point D and A.

Step 4: Join A and D.

Note: A rhombus is a special square where diagonals bisect each other perpendicularly.

A rough diagram for rhombus is given.

Step 1:

Draw the base for the diagram by using one of the measurements of 5.2 cm.

Step 2: Using point A and C, we will draw perpendicular bisector and let it intersect on the given line segment at point O.

Step 3: Taking point O as a center we draw arcs on the line EOD taking 3.2 as radius and intersect it.

[Since we have seen in the figure diagonals bisect each other at point O. So, BO= 3.2 cm= OD. ]

Step 4: Join B, BC, CD and AD.

Note: All the opposites’ sides of the rectangle are equal and parallel to each other, and all the sides are perpendicular to each other.

A rough diagram is shown.

Step 1: Draw a line segment of 5 cm .

Step 2: At point, A and B draw angle of 90 .

Step 3: Taking A and B as a center we draw an arc of 4 cm each.

Step 4: Joining the CD.

A rough sketch of this parallelogram:

(a) Draw a line segment OK of 5.5 cm and a ray at point K at a convenient angle

(b) Draw a ray at point O parallel to the ray at K
For parallel line.
i) Extend KO.
ii) Put compass needle on K and with some suitable radius draw an arc cutting KO and KA.
iii)With same radius draw another arc cutting extended KO by putting compass needle on O.
iv) Now put compass needle and pencil on cuts made by first arc on KO and KA. take it as new radius.
v) With same radius put compass needle on cut made by previous arc on extended KO and draw another arc cutting previous arc.
vi) Draw a ray from O passing this intersection point of arcs.
vii) With radius= 4.2 cm in compass draw arc cutting new ray and mark this point as Y.

The vertices A and Y are 4.2 cm away from the vertices K and O respectively so cut line segments KA and OY, each of 4.2 cm

(c) Join Y to A

OKAY is the required parallelogram