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11. In an isosceles right-angled triangle, the length of one leg is 10 metres. Find its area and perimeter.
  A.  50 m2; 34.14 m
  B.  30 m2; 36.15 m
  C.  45 m2; 35 m
  D.  55 m2; 30.25 m
     
   
View Answer

Shortcut:
Area of triangle =
1 / 2
x 10 x 10 = 50 sq m

Perimetre = AC + CB + AB
AB = √(102 + 102) = √200 = 10√2
∴ Perimetre = 10 + 10 +10√2 = 20 + 14.14 = 34.14 m

12. The perimeter of an isosceles triangle is 120 cm. If the base is 60 cm, find the lenght of equal sides.
  A.  25 cm
  B.  30 cm
  C.  25 cm
  D.  40 cm
     
   
View Answer

Shortcut:
The perimeter of an isosceles triangle is given as 'Z' cm. Now consider the following cases.
Case I: If the base o the isosceles triangle is given by 'm' cm, then the length of the equal sides is
Z − m / 2
cm
Case II: If the length of equal sides is given by 'a' cm, then the length of the base is (Z − 2a) cm.
Here, we will use case I
Z = 120, m = 60
Length of equal sides =
120 − 60 / 2
= 30 cm

13. The perimeter of an isosceles triangle is 100 cm. If the length of the equal sides is given by 32 cm, find the length of the base.
  A.  30 cm
  B.  36 cm
  C.  40 cm
  D.  25 cm
     
   
View Answer

Shortcut:
The perimeter of an isosceles triangle is given as 'Z' cm. Now consider the following cases.
Case I: If the base o the isosceles triangle is given by 'm' cm, then the length of the equal sides is
Z − m / 2
cm
Case II: If the length of equal sides is given by 'a' cm, then the length of the base is (Z − 2a) cm.
Here, we will use case II
Z = 100, a = 32
Length of base = 100 − (2 x 32) = 100 − 64 = 36 cm

14. Length of the side of an equilateral triangle is 4 √3 cm. Find its height.
  A.  6 cm
  B.  8 cm
  C.  3 cm
  D.  5 cm
     
   
View Answer

Shortcut:
To find the height of the equilateral triangle when the length of its side is given.
Height of the equilateral triangle =
√3 / 2
x side
Here, side = 4√3
Height of the equilateral triangle =
√3 / 2
x 4√3 = 6 cm

15. Height of an equilateral triangle is 6 cm. Find its side.
  A.  13√ 5 sq cm
  B.  12√ 3 sq cm
  C.  9√ 4 sq cm
  D.  11√ 7 sq cm
     
   
View Answer

Shortcut:
To find the area of an equilateral triangle when the height (h) is given.
Area of the equilateral triangle =
(h)2 / √3

Here, Height = 6
Area of the equilateral triangle =
(6)2 / √3

=
36 / √3
= 12√3
Hence, the area of equilateral triangle is 12√3 sq cm.

16. Perimeter of a square and an equilateral triangle is equal. If the diagonal of the square is 12 √2 cm, then find the area of the equilateral triangle.
  A.  64√3 sq cm
  B.  65√4 sq cm
  C.  58√5 sq cm
  D.  68√7 sq cm
     
   
View Answer

Shortcut:
The perimeter of a square is equal to the perimeter of an equilateral triangle. If the diagonal of the square is 'd' units, then
(i) the side of the square =
d / √2
units,
(ii) the side of the equilateral triangle =
4d / 3√2
units,
(iii) the area of the square =
d2 / 2
sq units, and
(iv) the area of the equilateral triangle =
2 / 3√3
x d2 sq units.
Here, d = 12√2
Area of the equilateral triangle =
2 / 3√3
x (12√2)2
=
2 / 3√3
x 12 x 12 x 2 = 64√3
Hence, the area of equilateral triangle is 64√3 sq cm.

17. The width of a rectangular hall is
3 / 4
of its length. If the area of the hall is 300 sq m, then the difference between its length and width is:
  A.  4 m
  B.  1 m
  C.  3 m
  D.  5 m
     
   
View Answer

Shortcut:
The area of rectangle is given by:
Area = Length x Breadth
Here, Let length = l metres. Then breadth =
3 / 4
l metres
∴ l x
3 / 4
l = 300
⇒ l2 =
300 x 4 / 3
= 400
or, l = 20 Hence, Length = 20 metres. So, Breadth =
3 / 4
x 20 = 15 metres
Hence, the difference between Length & Breadth = l − b = 20 − 15 = 5 metres

18. A room 8 x 6 m is to be carpeted by a carpet 2 m wide. The length of carpet required is:
  A.  20 m
  B.  25 m
  C.  24 m
  D.  28 m
     
   
View Answer

Shortcut:
The area of rectangle is given by:
Area = Length (l) x Breadth (b)
Here, Area of room = 8 x 6 = 48 sq m.
Breadth of carpet (b) = 2 m.
Using these values in the shortcut, we get:
Area = l x b
48 = l x 2
l =
48 / 2
= 24
Hence, the Length is 24 metres.

19. The length of a plot of land is 4 times its breadth. A playground measuing 1200 sq m occupies one-third of the total area of the plot. What is the length of the plot, in metres?
  A.  110 metres
  B.  120 metres
  C.  125 metres
  D.  130 metres
     
   
View Answer

Shortcut:
The area of rectangle is given by:
Area = Length (l) x Breadth (b)
Here, Breadth = b
Length = 4 x Breadth or, l = 4b
Area of plot = 4b x b
1 / 3
x Area = 1200
So, Area = 1200 x 3 = 3600
Using these values in the shortcut, we get:
Area = l x b
3600 = 4b x b
900 = b2
or, b = 30
So, length is 4b = 4 x 30 = 120
Hence, the Length is 120 metres.

20. If only the length of rectangular plot is reduced to
2 / 3
rd of its original lenght, the ratio of original area to reduced area is:
  A.  3:2
  B.  2:5
  C.  4:5
  D.  5:3
     
   
View Answer

Shortcut:
The area of rectangle is given by:
Area = Length (l) x Breadth (b)
Here, Length = l , Breadth = b
New Length =
2 / 3
l and new Breadth = b
So,
Original area / New Area
=
l x b / (2/3)l x b
=
3 / 2

Hence, the ratio is 3 : 2.

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