Similarity on Reduction Transformation – Definition, Questions | Reduction Factor, Center of Reduction

Transformation is a process of changing the image actual shape by rotation, reflection, or translation method. The images before and after transformation are similar to each other. Reduction transformation is nothing but reducing the size of the image based on similarity. Check the process and definition of reduction transformation along with solved questions on this page.

What is Reduction Transformation?

Transformation is where the shape or size of the preimage is not changed and non-rigid means where the size is changed but the shape remains the same. Reduction transformation means a figure undergoes reduction by a scale factor, such that the resulting figure is similar to the original figure.

In reduction transformation, the scale factor is also known as the reduction factor. It is the ratio of the dimensions of the pre-image to the original image.

Also, Check

Similarity on Reduction Transformation

We are giving the details about the similarity on reduction transformation with examples.

Here, ∆X’Y’Z’ is the reduced image of ∆XYZ

Reduction Transformation 1

The two triangles are similar as their corresponding angles are equal.

\(\frac { X’Y’ }{ XY } \) = \(\frac { Y’Z’ }{ YZ } \) = \(\frac { Z’X’ }{ ZX } \) = k

Here,

  • k is the reduction factor
  • P is the centre of reduction

In the above ∆XYZ undergoes reduction by a reduction factor of k and the resulting figure, original figures are similar to each other. So, the image remains constant in shape.

Examples of Transformation Reduction

Problem 1:
A triangle ABC is reduced to A’B’C’ and their areas are 150 sq cm and 20 sq cm respectively. If the perimeter of A’B’C’ is 25 cm, then find the perimeter of ABC.

Solution:
Let \(\frac { Area of trinagle A’B’C’ }{ area of triangle ABC } \) = k²
So, \(\frac { 20 }{ 150 } \) = k²
k² = \(\frac { 2 }{ 15 } \)
k = 0.365
Now, \(\frac { Perimeter of trinagle A’B’C’ }{ perimeter of triangle ABC } \) = k
\(\frac { 25 }{ perimeter of triangle ABC } \) = 20.365
perimeter of triangle ABC = 0.365 x 25 = 9.128 cm

Problem 2:
A rectangle PQRS has been reduced to a rectangle P’ Q’ R’ S’ and their perimeters are 18 cm and 6 cm respectively. If Q’R’ is 3 cm, then find QR.

Solution:
Let \(\frac { Perimeter of rectangle P’Q’R’S’ }{ perimeter of rectangle PQRS } \) = k
\(\frac { 6 }{ 18 } \) = k
k = 0.33
Now, \(\frac { Q’R’ }{ QR } \) = k
\(\frac { 3 }{ QR } \) = 0.33
QR = 3 x 0.33 = 1 cm

FAQ’s on Reduction Factor

1. What is reduction factor?

Reduction is nothing but reducing the size of the image. The reduction factor is the ratio of the reduced image size to the original image size in transformation. It is popularly known as a scale factor.

2. Define reduction transformation?

The process of decreasing the size of an image without making changes in its shape is called reduction transformation.

3. What is the difference between reduction and enlargement?

Both reduction or enlargement involves a change in the size of an object. An enlargement gives the same product with a larger proportion than the original. A reduction give proportionally smaller than the original.

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