Question:One side  of a rectangular field is 15 m and one of its diagonals is 17 m. Find the area of the field. 

Other side` = sqrt((17))^2 - ((15))^2`

              `= sqrt(289 - 225)`

              `= sqrt(64) = 8 m.`

:. Area` = (15 xx 8) sq.m = 120 sq.m.`

Question:Find the area of a triangle whose sides measure 13 cm, 14 cm and 15 cm. 

Let a = 13, b = 14 and C = 15.

Then, s `= 1/2 (a + b + c) = 21`

:. (s - a) = 8, (s - b)

= 7 and (s - c) = 6

:. Area` = sqrt(s (s - a) (s - b) (s - c)`

  `= sqrt(21 xx 8 xx 7 xx 6)`

   `= 84 cm^2`

Question:Find the area of a square, one of whose diagonal is 3.8 m long. 

Area of the square` = 1/2 xx (diagonal)^2`

  `= (1/2 xx 3.8 xx 3.8) m^2 = 7.22 m^2`

Question:Find the area of an equilateral triangle each of whose sides is 8 cm long. 

Area of the triangle` = sqrt(sqrt(3)/4 xx 8 xx 8) cm^2`

                   `= 16sqrt(3)  cm^2`

Question:Find the area of a right angle triangle whose base is 12 cm and hypotenuse 13 cm. 

Height of the triangle` = sqrt((13))^2 - (12)^2`

            `= sqrt(25) = 5 cm.`

:. Its area` = 1/2 xx Base xx Height`

      `= ( 1/2 xx 12 xx 5) cm^2`

      `= 30  cm^2`

Question:The base of a triangular field is three times its altitude. If the cost of cultivating the field at Rs. 24.68 per hectare be Rs. 333.18, find its base and height. 

Area of the field` = (Total cost)/(Rate)`

                     `= ((333.18)/(24.68))` hectares

                     = 13.5 hectares

  
     `= (13.5 xx 10000) m^2`

     `= 135000 m^2`

Let, altitude = x metres and base = 3x metres.

Then,` 1/2 xx 3x xx x = 135000 or x^2`

  `= 90000 or x = 300.`

:. Base = 900 m & Altitude = 300 m.

Question:Find the area of a rhombus one side of which measure 20 cm and one diagonal 24 cm. 

Let, other diagonal = 2x cm.

Since halves of diagonal and one side of a rhombus form a right angle triangle with  

side as hypotenuse, we have:

`(20)^2 = (12)^2 + x^2 or x = sqrt((20))^2 - (12)^2`

`= sqrt(256) = 16 cm.`

:. Other diagonal = 32 cm.

:. Area of the rhombus` = 1/2 xx` (product of diagonals)

`= (1/2 xx 24 xx 32) cm^2 = 384 cm^2`

Question:A lawn is in the form of a rectangle having its sides in the ratio 2 : 3. The area of the lawn is `1/6` hectares. Find the length and breadth of the lawn. 

Let, length = 2x metres and breadth = 3x metres.

Now, area` = (1/6 xx 10000) m^2`

           `= (5000/3) m^2`

`:. 2x xx 3x = (5000)/3 or x^2 = 2500/9 or x = (50/3) m.`

:. Length` = 2x = (100)/(3) m`

        `= 33 1/3  m & Breadth = 3x = (3 xx 50/3)` m

       `= 50 m.`

Question:A room is half as long again as its is broad. The cost of carpeting the room at Rs.5 per sq.m. is Rs.270 and the of papering the four walls at Rs.10 per `m^2` is Rs. 1720. If a door and 2 windows occupy 8 sq. metres, find the dimensions of the room. 

Let the breadth = x metres & length` = (3x)/2` metres.

Area of the floor` = ((Total cost of carpeting)/(Rate|m^2)) m^2`

                    `= (270/5) m^2`

                   `= 54 m^2`


`:. x xx (3x)/2 = 54 or x^2`

`= 54 xx 2/3 = 36 or x = 6`

:. Breadth = 6 m & length` = (3/2 xx 6) m`

                   `= 9 m`

Now, papered area` = (1720/10) m^2`

                     `= 172 m^2`

Area of 1 door and 2 windows` = 8m^2`

Total area of 4 walls` = (172 + 8) m^2`

                  `= 180 m^2`

`:. 2 (length + breadth) xx height = 180`

`or 2 (9 + 6) xx H = 180 or H = (180/30)`

`= 6 m`

Question:A rectangular grassy plot 110 m by 65 m has a gravel path 2.5 m wide all round it on the inside. Find the cost of gravelling the path at 80 paise per sq. metre. 

Area of the plot` = (110 xx 65) m^2`

                      `= 7150 m^2`

Area of the plot excluding the path

`= [(110 - 5) xx (65 - 5)] m^2`

`= 6300 m^2`

:. Area of the path` = (7150 - 6300) m^2`

                   `= 850 m^2`

:. Cost of gravelling the path` = Rs. (850 xx 80/100)`

                 `= Rs. 680`