# Reduction Transformation

We will discuss here about the similarity on Reduction transformation.

In the figure given below ∆X’Y’Z’ is a reduced image of ∆XYZ.

The two triangles are similar. Here also the triangles are equiangular and $$\frac{X’Y’}{XY}$$ = $$\frac{Y’Z’}{YZ}$$ = $$\frac{Z’X’}{ZX}$$ = k.

Here k is known as the reduction factor and P is known as the centre of reduction.

Therefore, in a size transformation, a given figure undergoes enlargement or reduction by a scale factor k, such that the resulting figure is similar to the original figure, i.e., the image retains the shape of the original object.

If ∆XYZ is transformed to ∆X’Y’Z’ by a scale factor k about the point P, we get $$\frac{PX’}{PX}$$ = $$\frac{PY’}{PY}$$ = $$\frac{PZ’}{PZ}$$ = k.

2. A rectangle PQRS has been reduced to a rectangle P’ Q’ R’ S’ and their areas are 192 cm$$^{2}$$ and 12 cm$$^{2}$$ respectively. If Q’ R’ is 3 cm, then find QR.

Solution:

Let $$\frac{area of P’ Q’ R’ S’}{area of PQRS}$$ = k$$^{2}$$

Therefore, $$\frac{12 cm\(^{2}$$}{192 cm$$^{2}$$}\) = k$$^{2}$$

⟹ $$\frac{1}{16}$$ k$$^{2}$$

⟹ k = $$\frac{1}{4}$$

Now, $$\frac{Q’ R’}{QR}$$ = k

⟹ $$\frac{3 cm}{QR}$$ = $$\frac{1}{4}$$

⟹ QR = 12 cm

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