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

9th Grade Math

From Reduction Transformation to HOME PAGE