If the sides of a rectangle are multiplied by k, a similar rectangle is obtained.
The new area = ka × kb
= k^{2} ab
= k^{2} × the old area
If an object or figure is enlarged by a scale factor of k, then
the area of the image = k^{2} × the area of the object
Consider the following similar shapes. Find
i) the scale factor
ii) the length or area marked by the unknown.
Problem 1 :
Solution :
i) scale factor :
10 = k^{2} × 40
k^{2} = 10/40
k^{2} = 1/4
k = 1/2
ii) the length :
Length of smaller figure = Scale factor x length of small figure
x = k × 10
x = 1/2 × 10
x = 5 cm
Problem 2 :
Solution :
i) scale factor :
9 = k × 6
k = 9/6
k = 3/2
Here, we find area of the large figure.
ii) Area :
Area of large shape = k^{2} × Area of smaller shape
x^{2} = k^{2} × 8
By applying the value of k, we get
x^{2} = (3/2)^{2} × 8
= 18 cm^{2}
Area of the large shape is 18 cm^{2}.
Problem 3 :
Solution :
i) scale factor :
5 = k × 2
k = 5/2
ii) Area :
Area of large shape = k^{2} × Area of smaller shape
25 = (5/2)^{2} × x
25 = 25x/4
x = 25(4/25)
x = 4 cm^{2}
Problem 4 :
Solution :
i) scale factor :
6 = k^{2} × 20
k^{2} = 6/20
k^{2} = 3/10
k = √3/10
k = √0.3
k ≈ 0.548
ii) the length :
Length of smaller shape = k (length of larger shape)
= k × 8
= 0.548 × 8
≈ 4.38 cm
Problem 5 :
Solution :
i) scale factor :
31.8 = k^{2 }× 55.4
k^{2 }= 31.8/55.4
k^{2} = 0.574
k = √0.574
k ≈ 0.758
ii) the length or area :
x = k × 4.2
x = 0.758 × 4.2
x ≈ 3.18 m
Problem 6 :
Solution :
i) scale factor :
7.25 = k × 5
k^{ }= 7.25/5
k ≈ 1.45
ii) Area :
Area of the large shape = k^{2} x area of small shape
70 = k^{2} (x)
Applying the value of k, we get
70 = (1.45)^{2} x
x = 70/(1.45)^{2}
x = 70/2.1025
x = 33.29
Approximately 33.3 cm^{2}
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM