Let us consider the following square.
Problem 1 :
Find the area of a square with a diagonal of 3 cm.
Solution :
Diagonal = 3 cm
Diagonal d = √2 a
√2 a = 3
a = 3/√2
a = 3/1.414
a = 2.121
Area of square = a²
= (2.121)²
= 4.5 cm²
So, area of a square is 4.5 cm².
Problem 2:
Find the area of a square whose diagonal is 9 cm.
a) 18 cm² b) 81/2 cm² c) 36 cm² d) 81 cm²
Solution :
Given, diagonal = 9 cm
Diagonal d = √2 a
√2 a = 9
a = 9/√2
Area of square = a²
= (9/√2)²
= 81/2 cm²
So, option (b) is correct.
Problem 3:
The area of a square is 2√2 + 3. What is the length of a side of the square?
a) √2 - 1 b) 2√2 - 1 c) √2 + 1 d) 2√2 + 1
Solution :
Area of square = 2√2 + 3
a^{2} = 2√2 + 3
So, side length of the square is √2 + 1.
Problem 4 :
The area of the square ABCD is 9/2 cm². Find the length of BD.
Solution :
Given, area of a square = 9/2 cm²
Area = a²
9/2 = a²
a = √ (9/2)
a = 3/√2 cm
a = 3/1.414
a = 2.12 cm
So, length of a square is 2.12 cm.
Problem 5 :
Find the area of a square one of whose diagonal is 3.8 m long.
Solution :
Diagonal d = √2 a
√2 a = 3.8
a = 3.8/√2
a = 3.8/1.414
a = 2.687
Area of square = a²
= (2.687)²
= 7.22 m²
So, area of a square is 7.22 m².
Problem 6:
The diagonals of two squares are in the ratio 2 : 5. Find the ratio of their areas.
Solution :
Let the diagonals of first and second square be 2x and 5x respectively.
√2 a = 2x a = 2x/√2 a = √2x Area of first square = a² = (√2x)² = 2x² |
√2 a = 5x a = 5x/√2 Area of first square = a² = (5x/√2)² = 25x²/2 |
Ratio of area = 2x²: 25x²/2
= 2 : 25/2
= 4 : 25
Therefore the ratio of the areas of two squares is 4 : 25.
Problem 7 :
A man walked diagonally across a square lot. Approximately, what was the percent saved by not walking along the edges?
a) 20 b) 24 c) 30 d) 33
Solution :
Diagonal of square = √2 a
= 1.414 a
When you walk on two sides of square = 2a
When you walk diagonally = 1.414 a
Percentage saving = [(2a - 1.414a)/2a] × 100
= 29.5%
Therefore, percentage saving is 29.5% (approx)
So, option (c) is correct.
Problem 8 :
A man walking at the speed of 4 kmph crosses square field diagonally in 3 minutes. The area of the field is.
a) 18000 m² b) 19000 m² c) 20000 m² d) 25000 m²
Solution :
Speed of man = 4 km/hr
= 4000/60 m per min
Distance covered in 3 minutes = 3 × 4000/60
= 200 meter
Diagonal of square = 200 meter
Area of square = 1/2 × (diagonal)²
= 1/2 × 200 × 200
= 20000 square meter
So, option (c) is correct.
Problem 9 :
If the length of the diagonal of a square is 20 cm, then its perimeter must be
a) 10√2 cm b) 40 cm c) 40√2 cm d) 200 cm
Solution :
Diagonal of a square = √2 a
Side = diagonal/√2
= 20/√2 cm
= 20/√2 × √2/√2
= 10√2 cm
Perimeter of square = 4 a
= 4 × 10√2
= 40√2 cm
So, option (c) is correct.
Problem 10 :
The area of a square field is 69696 cm². its diagonal will equal to.
a) 313.296 m b) 353.596 m
c) 373.296 m d) 393.296 m
Solution :
The area of a square = 69696 cm²
Area = a²
a = √69696
a = 264 cm
Diagonal = √2 a
= √2 × 264
= 1.414 × 264
= 373.296 m
So, option (c) is correct.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM