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•A simple closed curve made up of only line segments is called a polygon. • A diagonal of a polygon is a line segment connecting two non- consecutive vertices. •A convex polygon is a polygon in which no portion of its any diagonal is in its exterior. •A quadrilateral is a polygon having only four sides. • A regular polygon is a polygon whose all sides are equal and also all angles are equal. • The sum of interior angles of a polygon of n sides is (n-2) straight angles. •The sum of interior angles of a quadrilateral is 360°. • The sum of exterior angles, taken in an order, of a polygon is 360°. •Trapezium is a quadrilateral in which a pair of opposite sides isparallel. •Kite is a quadrilateral which has two pairs of equal consecutive sides. • A parallelogram is a quadrilateral in which each pair of opposite sides is parallel. •A rhombus is a parallelogram in which adjacent sides are equal. •A rectangle is a parallelogram in which one angle is of 90 0 •A square is a parallelogram in which adjacent sides are equal and one angle is of 90 0 • In a parallelogram, opposite sides are equal, opposite angles are equal and diagonals bisect each other. • In a rhombus diagonals intersect at right angles. • In a rectangle diagonals are equal. • Five measurements can determine a quadrilateral uniquely. • A quadrilateral can be constructed uniquely if the lengths of its four sides and a diagonal are given. • A quadrilateral can be constructed uniquely if the lengths of its three sides and two diagonals are given. • A quadrilateral can be constructed uniquely if its two adjacent sides and three angles are given. • A quadrilateral can be constructed uniquely if its three sides and two included angles are given. In examples 1 to 8, there are four options out of which one is correct.

Write the correct answer.

Example 1:The number of diagoynals in a polygon oyf n sides is (a) nn( 1) 2 (b) nn( 2) 2 (c) nn( 3) 2 (d) n (n-3).

Solution:The correct answer is (c).

Example 2:The angles of a quadrilateral ABCD taken in an order arein the ratio 3 : 7 : 6 : 4. Then ABCD is a

(a) kite(b)parallelogram (c) rhombus(d)trapezium

Solution:The correct answer is (d).

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l l or, 4l = 180° - l or, 5 l = 180° or, l = 180
5 So, ol = 36°

Number of sides of polygon =

360
exterior angle 360
36
= 10 of sides of the polygon. so, exterior angle = 180 0 - 108 0 = 72°

Number of sides =

0 0

360 360

5 exteriorangle 72 angle A = 110°. Since in a rhombus, all sides are equal so, PA = AI = IR =

RP = 6cm

Also, rhombus is a parallelogram

so, adjacent angle, I = 180° - 110° = 70°

Steps of construction

1.Draw AI = 6 cm

2.Draw ray

AX such that oIAX = 110° and draw IY such that oAIY = 70°.

3. With A and I as centres and radius 6cm draw arcs

intersecting AX and IY at P and R respectively.

4.Join PR.

Thus, PAIR is the required rhombus.

Find the angles of the rhombus.

equal to its side, that is, PQ = QR = RS = PS = PR So, pPRS and pPQR are equilateral. oS = oQ = 60° [Each angle of an equilateral triangle is 60°.] and oP = o1 + o2 = 60° + 60° = 120° = oR

Hence oS = oQ = 60° and oP = oR = 120°

G. If HG = 5

x + 1 and EG = 4x + 19, find x.

Since diagonals of a rectangle bisect each other,

HP = 2HG = 2 (5x + 1) = 10x +2

andOE = 2EG = 2(4lo+19) = 8lo+ 38

Diagonals of a rectangle are equal. So HP = OE

or 10l + 2 = 8l + 38 or 2l = 36 or l = 18 m,.RICE is a rhombus. Find lgongoqy

Justify your findings. Hence, find the

perimeter of the rhombus. mypmypnx n x

We have to find the values of lgongoq.

i.e. OE, OY and side IR of the rhombus and perimeter of the rhombus.

What do we know?

RICE is a rhombus and

OC = 12, OE = 5, OI = lo+ 2, OR = lo+on

OExmym

m,. (1)We have to find the parts of the diagonal. Use diagonals of a rhombus bisect each other. (2)We have to find the side of the rhombus. We use diagonals intersect at right angles and apply pythagoras theorem. (3)Since all sides of a rhombus are equal, perimeter of the rhombus = 4 × side.

Step 1. OI = OE , l + 2 = 5 or l = 5 - 2 = 3.

OC = OR , 12 = n + l or n = 12 - l

12 - 3 = 9

Step 2.EOR is a right triangleER

2 = OE 2 + OR 2 = 5 2 + 12 2 = 25 + 14 4 = 169 zl m p xl y x ER= 169
= 13cm

Step 3.Since all sides of a rhombus are equal.

n

RE = RI = IC = CE = 13 cm.

Perimeter of RICE = 4 × RE = 4 × 13 cm

= 52 cm

We have been asked yto find

x, y and z and we have found that. '"y, x + 2 = 5 and x = 3 g 3 + 2 = 5

Hence value of x is correct.

x + y = 12 x = 3 and y = 9 and 3 + 9 = 12 gyvalue of y is correct.

Perimeter of rhombus =

nn

2 d1 + d2

(where d1 and d2 are diagonals) 22

2 24 + 10

= 2 576+ 100 2 676 = 52 cm (i)If RICE is a parallelogram, not a rhombus can you find x, y and z ? (ii)If RICE is a rhombus with EC = 20 cm and OC = 12 cm, can you find x, y, z ? xl y m,. Construct a rhombus with side 4.5cm and diagonal 6cm. mypmypnx n x

What do you know?

Here, side of rhombus = 4.5 cm.

Diagonal of rhombus = 6 cm.

What do we need to make rhombus?

4 sides and its one diagonal

Plan a strategy

(1)Use property of rhombus- all sides are equal. (2)Make a free hand roughsketch and name itABCD. Solve

Step-1.Draw AB = 4.5 cm.

Step-2. With A as centre and radius

6 cm draw an arc above AB.

Step-3.With B as centre draw an

arc to cut the arc drawn in step 2 at pt C.

Step-4.Join AC and BC.

Step-5.With A and C as centreand radius 4.5 cm drawarcs to intersect each other at D.

Step-6.ABCD is required

rhombus.

Checking:

V erify your figure by adopting some other property of rhombus.

Step 1. Join BD to intersect AC as O.

Step 2. Measure AOB. Is it 90°?

Step 3. Measure OA and OC. Are they equal?

Step 4. Measure OB and OD. Are they equal?

If your answer to 2, 3, 4 is yes it means what

you have cons tructed is a right angle.

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(a) P(b) Q(c) R(d) S

7.Which of the following figures satisfy the following property?

- Has two pairs of congruent adjacent sides. (a) P(b) Q (c) R(d) S

8.Which of the following figures satisfy the following property?

- Only one pair of sides are parallel. (a) P(b) Q(c) R(d) S

9.Which of the following figures do not satisfy any of the following

properties? - All sides are equal. - All angles are right angles. - Opposite sides are parallel. (a) P(b) Q(c) R(d) S

10. Which of the following properties describe a trapezium?

(a) A pair of opposite sides is parallel. (b) The diagonals bisect each other. (c) The diagonals are perpendicular to each other. (d) The diagonals are equal.

11.Which of the following is a property of a parallelogram?(a) Opposite sides are parallel.(b) The diagonals bisect each other at right angles.

(c) The diagonals are perpendicular to each other. (d) All angles are equal.quotesdbs_dbs44.pdfusesText_44

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