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(iv) 4 m

Area = Base x height = 96 

⇒ height = \(\frac{96}{24}\) = 4

(ii) 32 sq. cm

Area = \(\frac{1}{2}\)(d₁ x d₂) = \(\frac{1}{2}\) x 8 x 8 = 32

(iii) 12 sq. cm

Area = Base x Height = 4 x 3 = 12 cm²

(a) Area of the rectangle = (base x height) sq. units 

Base = 5 units 

Height = 5 units 

∴ Area = (5 x 5) = sq. units = 25 sq. units

(b) Area of the rectangle = (base x height) sq. units 

Base = 4 units 

Height = 1 units 

∴ Area = (4 x 1) = sq. units = 4 sq. units

(c) Area of the rectangle = (base x height) sq. units 

Base = 2 units 

Height = 3 units 

∴ Area = (2 x 3) = sq. units = 6 sq. units

(d) Area of the rectangle = (base x height) sq. units 

Base = 4 units 

Height = 4 units 

∴ Area = (4 x 4) = sq. units = 16 sq. units

(e) Area of the parallelogram = (base x height) sq. units 

Base = 7 units 

Height = 5 units 

= 7 x 5 = 35 sq. units

Yes, we can draw a rhombus with one of its diagonals equal to its side length. In such case the diagonal will divide the rhombus into two congruent equilateral triangles.

Given perimeter of the rhombus = 180 cm 

∴ One side of the rhombus = \(\frac{180}{4}\) = 45 cm 

Given area of the rhombus = 315 cm² 

b x h = 315 

45 x h = 315 = \(\frac{315}{45}\)

h = 7 cm 

Altitude of the rhombus = 7 cm

In a square

(i) All sides are equal. 

(ii) Opposite sides are parallel 

(iii) Diagonals divides the square into 4 right angled triangles of equal area 

(iv) The diagonals bisect each other at right angles. 

So it become a rhombus also. 

But in a rhombus 

(i) Each angle need not equal to 90°. 

(ii) The length of the diagonals need not be equal. Therefore it does not become a square.

If two parallel sides are equal in length. Then it can be divided into two equal triangles.

Length of the parallel sides a = 200 cm 

b = 150 cm 

Height h = 50 cm

Area of the trapezium = \(\frac{1}{2}\) x h(a + b) sq. units

= \(\frac{1}{2}\) x 50 (200+ 150) cm²

= \(\frac{1}{2}\) x 50 x 350 cm² = 8750 cm²

Cost for 10 sq. cm glass = Rs 68

∴ Cost of 8750 cm² glass = \(\frac{8750}{10}\) x 6 = Rs 5250

Cost of glass used = Rs 5,250

(ii) 8 cm

\(\frac{1}{2}\) x d₁ x d₂ = 128 

⇒ d₂ = \(\frac{128\,\times\,2}{32}\) = 8cm

Parallel sides of the trapezium a = 42 m; b = 36 m 

Also height h = 30 m 

Area of the trapezium = \(\frac{1}{2}\) x h x (a + b) sq. unit 

= \(\frac{1}{2}\) x 30 x (42 + 36) m²

= \(\frac{1}{2}\) x 30 x 78 m²

Area = 1,170 m²

Cost of levelling 1 m² = Rs 135

∴ Cost of levelling 1170 m² = Rs 1170 x 135 = Rs 1,57,950

Cost of levelling the ground = Rs 1,57,950

Let one diagonal of the rhombus = d₂ cm 

The other diagonal d₂ = \(\frac{1}{2}\) x d₁ cm 

Area of the rhombus = 576 sq. cm 

\(\frac{1}{2}\) x (d₁ x d₂) = 576 

\(\frac{1}{2}\) x (d₁ x \(\frac{1}{2}\)d₁) = 576 

d₁ x d₁ = 576 x 2 x 2 = 6 x 6 x 4 x 4 x 2 x 2

d₁ x d₁ = 6 x 4 x 2 x 6 x 4 x 2 

d₁ = 6 x 4 x 2 

d₁ = 48 cm 

d₂ = \(\frac{1}{2}\) x 48 = 24 cm 

∴ Length of the diagonals d₁ = 48 cm and d₂ = 24 cm.

(i) (a) PQ and RS (b) QR and PS 

(ii) (a) PQ and QR (b) PS and RS 

(iii) (a) PR and Question are diagonals.

Let one of the diagonals of rhombus be ‘d₁’ cm and the other be d₂ cm.

Give d₁ = 3 x d₂ 

Also d₁ + d₂ = 24 cm 

⇒ 3d₂ + d₂ = 24 

4d₂ = 24 

d₂ = \(\frac{24}{4}\)

d₂ = 6 cm 

d₁ = 3 x d₂ = 3 x 6 

d₁ = 18 cm 

∴ Area of the rhombus = \(\frac{1}{2}\) x d₁ x d₂ sq. units 

= \(\frac{1}{2}\) x 18 x 6 cm² = 54 cm² 

Area of the rhombus = 54 cm²

(iii) 90°

Angles of a rhombus bisect at right angles.

Given the length of one diagonal d₁ = 8 cm; 

Area of the rhombus = 100 sq. cm 

\(\frac{1}{2}\)(d₁ x d₂) = 100 

\(\frac{1}{2}\) x 8 x d₂ = 100 

8 x d₂ = 100 x 2 

d₂ = \(\frac{100\,\times\,2}{8}\) = 25 cm 

Length of the other diagonal d₂ = 25 cm

(iii) 13 cm

Base = \(\frac{area}{height}=\frac{52sq.cm}{4cm}\) = 13 cm

(i) 45 sq. cm

\(\frac{1}{2}\) x h x (a + b) = \(\frac{1}{2}\) x 5 x (10 + 8) = 45

We know that in a parallelogram opposite sides are equal.

∴ 3x = 18 

x = \(\frac{18}{3}\)

x = 6 and 

3y – 1 = 26 

3y = 26 + 1 = 27 

y = \(\frac{27}{9}\)

y = 3

(i) 70 sq. m 

= base x height = 10m x 7m = 70 sq.m

Area of the parallelogram = 1470 sq. cm 

Considering AB = base = 49 cm

Height = DF 

Area = base x height 

49 x DF = 1470 

DF = \(\frac{1470}{49}\)

DF = 30 cm 

Now considering AD as base 

Base = AD = 35 cm; height = BE

Base x Height = 1470 

35 x BE = 1470 : BE = \(\frac{1470}{35}\)

BE = 42 cm; DF = 30 cm; BE = 42 cm

Area of each title = \(\frac{1}{2}\) x d₁ x d₂ sq. units 

= \(\frac{1}{2}\) x 40 x 25 cm = 500 cm 

∴ Area of 2000 titles = 500 x 20,000 = 10,000 cm 

= 100 m²

Cost of polishing 1 m² = Rs 5

∴ Cost of polishing 100 m² = 5 x 100 = Rs 500

(i) Given the diagonals d₁ = 16 cm; d₂ = 8 cm 

Area of the rhombus = \(\frac{1}{2}\)(d₁ x d₂) sq. units 

= \(\frac{1}{2}\) x 16 x 8 cm² = 64 cm² 

Area of the rhombus = 64 cm² 

(ii) Given base b = 15 cm; Height h = 11 cm 

Area of the rhombus = (base x height) sq. units 

= 15 x 11 cm² = 165 cm² 

Area of the rhombus = 165 cm²

(i) Glass of a car windows. 

(ii) Eye glass (glass in spectacles) 

(iii) Some bridge supports. 

(iv) Sides of handbags.

(i) Given base b = 11 cm; height h = 3 cm 

Area of the parallelogram = b x h sq. units = 11 x 3 cm = 33 cm 

Also perimeter of a parallelogram = Sum of 4 sides 

= 11 cm + 4 cm + 11 cm + 4 cm = 30 cm 

Area = 33 cm²; Perimeter = 30 cm.

(ii) Given base b = 7 cm 

height h = 10 cm 

Area of the parallelogram = b x h sq. units 

= 7 x 10 cm² = 70 cm² 

Perimeter = Sum of four sides 

= 13 cm + 7 cm + 13 cm + 7 cm = 40 cm 

Area = 70 cm², Perimeter = 40 cm

Given base 6 = 21 cm 

Area of parallelogram = 735 sq. cm 

b x h = 735 

21 x h = 735 

h = \(\frac{735}{21}\)

h = 35 cm 

∴ Height of the trophy = 35 cm

Given the parallel sides a = 105 cm; b = 50 cm; 

Height = 60 cm 

Area of the trapezium = \(\frac{1}{2}\) x h x (a + b) sq. units 

= \(\frac{1}{2}\) x 60 x (105 + 50) cm²

= 30 x 155 cm² = 4650 cm² 

For 100 cm² cost of glass used = Rs 15 

∴ For 4650 cm² cost of glass = Rs \(\frac{4650}{100}\) x 15 = Rs 697.50 

Cost of the glass used = Rs 697.50

Given the parallel sides a = 81 cm; b = 64 cm 

Distance between ‘a’ and ‘b’ is height h = 6 cm 

Area of the trapezium = \(\frac{1}{2}\) x h(a + b) sq. units 

= \(\frac{1}{2}\) x 6 x (81 + 64) = 3 x 145 cm² = 435 cm² 

Cost of painting 1 cm² = Rs 2 

Cost of painting 435 cm² = Rs 435 x 2 = Rs 870 

Cost of painting = Rs 870 

Given side of the square is 48 m

Area of the square = (side x side) sq. unit = 48 x 48 m²

Height of the parallelogram = 18 m

Area of the parallelogram = ‘bh’ sq. units = b x 18 m²

Also area of the parallelogram = Area of the square

b x 18 = 48 x 48

b = \(\frac{48\,\times\,48}{18}\) = 8 x 16 = 128 m

Base of the parallelogram = 128 

(i) Given Height h = 10 m; Parallel sides a = 12 m; b = 20 m 

Area of the Trapezium = \(\frac{1}{2}\)h(a + b) sq. units 

= \(\frac{1}{2}\) x 10 x (12 + 20)m² 

= (5 x 32)m² = 160 m²

(ii) Given the parallel sides a = 13 cm; 6 = 28 cm 

Area of the trapezium = 492 sq. cm 

\(\frac{1}{2}\)h(a + b) = 492 

\(\frac{1}{2}\) x h x (13 + 28) = 492 

h x 41 = 492 x 2

h = \(\frac{492\,\times\,2}{41}\)

h = 24 cm

(iii) Given height ‘h’ = 19 m; Parallel sides b = 16 m 

Area of the trapezium = 323 sq. m 

\(\frac{1}{2}\)h(a + b) = 323 

\(\frac{1}{2}\) x h x (a + 16) = 323 

a + 16 = \(\frac{323\,\times\,2}{19}\) = 34 

a = 34 – 16 = 18 m 

a = 18 m

(iv) Given the height h - 16 cm; Parallel sides a = 15 cm 

Area of the trapezium = 360 sq. cm 

\(\frac{1}{2}\) x h x (a + b) = 360 

\(\frac{1}{2}\) x 16 x (15 + 6) = 360 

15 + b = \(\frac{360}{8}\) = 45 

b = 45 – 15 = 30 

b = 30 cm

Tabulating the results we get

(i) Given diagonal d₁ = 19 cm; d₂ = 16 cm 

Area of the rhombus = \(\frac{1}{2}\)(d₁ x d₂) sq. units = \(\frac{1}{2}\) x 19 x 16

= 152 cm²

(ii) Given diagonal d₁ = 26 m; Area of the rhombus = 468 sq. m 

= \(\frac{1}{2}\)(d₁ x d₂) = 468; (26 x d₂) = 468 x 2 

d₂ = \(\frac{468\,\times\,2}{26}\) = d₂ = 36 m

(iii) Given diagonal d₂ = 12 mm; Area of the rhombus 

= 180 sq. m 

\(\frac{1}{2}\)(d₁ x d₂) = 180 

\(\frac{1}{2}\)(d₁ x 12) = 180

d₁ x 12 = 180 x 2 

d₁ = \(\frac{180\,\times\,2}{12}\)

d₁ = 30 mm 

Diagonal d₁ = 30 mm

Tabulating the results we have

(i) Given Base 6 = 18 cm; Height h = 5 cm 

Area of the parallelogram = b x h sq. units 

= 18 x 5 cm² 

= 90 cm²

(ii) Base b = 8m; Area of the parallelogram = 56 sq. m 

b x h = 56 

8 x A = 56 

h = \(\frac{56}{8}\)

h = 7 m

(iii) Given Height h = 17 mm 

Area of the parallelogram = 221 sq. mm 

b x h = 221 

b x 17 = 221 

b = \(\frac{221}{17}\)

b = 13 m 

Tabulating the results, we get

(iv) 28 cm

Area = \(\frac{1}{2}\) x h x (a + b) = 140 = \(\frac{1}{2}\) x h x 10 

⇒ h = 28

(i) Given length l = 12 m; Breadth b = 8 cm 

∴ Area of rectangle = l x b sq. units = 12 x 8 m² 

= 96 m²

Perimeter of the rectangle = 2 x (1 + b) units 

= 2 x (12 + 8)m = 2 x 20 = 40m

(ii) Given Length l = 15 cm; Area of the rectangle = 90 sq. cm 

l x b = 90; 15 x 6 = 90; b = \(\frac{90}{15}\) = 6 cm 

Perimeter of the rectangle = 2 x (l + b) units = 2 x (15 + 6) cm = 2 x 21 cm 

= 42 cm

(iii) Given Breadth of rectangle = 50 mm; Perimeter of the rectangle = 300 mm

2 x (l + b) = 300

2 x (l + 50) = 300

l + 50 = \(\frac{300}{2}\) = 150

l = 150 – 50

l = 100

Area = l x b sq. untis = 100 x 50 mm² = 5000 mm²

(iv) Length of the rectangle = 12 cm; Perimeter = 44 cm

2(l + b) = 44

2(12 + b) = 44

12 + b = \(\frac{44}{2}\)

12 + b = 22; b = 22 – 12; b = 10 cm

Area = l x b sq. units

= 12 x 10 cm² = 120 cm²

(i) Area of the rhombus = \(\frac{1}{2}\)(d₁ + d₂) sq. units 

= \(\frac{1}{2}\)(11 + 13) sq. units

= \(\frac{1}{2}\) x (24) cm² = 12 cm²

(ii) Base = 10 cm; Height = 7 cm 

Area of the rhombus = b x h sq. units = 10 x 7 cm² 

= 70 cm²

When the diagonals of a rhombus become equal it become a square.

If all sides are given, then by adding all the four lengths we can find the perimeter of a trapezium.

If we know the length of one side we can find the perimeter using 4 x side units.

(i) Given side a = 60 cm 

Area of the square = a x a sq.units = 60 x 60 cm² 

= 3600 cm²

Perimeter of the square = 4 x a units = 4 x 60 cm 

= 240 cm

(ii) Given area of a square = 64 sq. m 

a x a = 64 

a x a = 8 x 8 

a = 8m 

Perimeter = 4 x a 

= 4 x 8 

= 32 m

(iii) Given perimeter of the square = 100 mm 

4 x a = 100

a = \(\frac{100}{4}\) mm 

a = 25 mm 

Area = a x a sq. units 

= 25 x 25 mm²

= 625 mm²

Answer is (B) system of units

Answer is (D) mole

Answer is (D) 10 N

Answer is (C) parallax

Answer is (C) quantity of matter contained in the body.