EXPLORING DILATIONS

A dilation is a transformation that moves each point on the original figure along a straight line drawn from a fixed point. The point is called the point of dilation and the distance moved is determined by the scale factor used. Dilations produce figures that are the same shape as the original, but not the same size. 

When the scale factor is greater than one, the dilation is called an enlargement. 

When the scale factor is less than one, the dilation is called a reduction. 

The toy models given below are scaled-down replicas of the Saturn V rocket that powered the moon flights. Each replica is a transformation called a dilation.

Unlike the other transformations we have studied — translations, rotations, and reflections — dilations change the size (but not the shape) of a figure.

Exploring Dilations 

Triangle P′Q′R′ is a dilation of triangle PQR. Point C is the center of dilation.

Step 1 :

Use a ruler to measure segments CP, CP', CQ, CQ', CR and CR' to the nearest millimeter. Record the measurements and ratios. 

CP  =  2.5 cm,  CP'  =  5 cm

CP'/CP  =  5/2.5  =  2

CQ  =  2 cm,  CQ'  =  4 cm

CQ'/CQ  =  4/2  =  2

CR  =  3 cm,  CR'  =  6 cm

CR'/CR  =  6/3  =  2

Step 2 :

Write a conjecture based on the ratios in the table.

All the ratios are equal and it is 2. The distances are proportional.

Step 3 :

Use a ruler to measure segments CP, CP', CQ, CQ', CR and CR' to the nearest millimeter. Record the measurements and ratios. 

PQ  =  1 cm,  P'Q'  =  2 cm

P'Q'/PQ  =  2/1  =  2

QR  =  1 cm,  Q'R'  =  2 cm

Q'R'/QR  =  2/1  =  2

PR  =  1.4 cm,  P'R'  =  2.8 cm

P'R'/PR  =  2.8/1.4  =  2

Step 4 :

Write a conjecture based on the ratios in the table.

All the ratios are equal and it is 2. The lengths are proportional.

Step 5 :

Measure the corresponding angles and describe your results.

The corresponding angles are congruent.

PQR and P'Q'R' are similar triangles.

Since the corresponding angles are congruent, it is clear that the dilations change size but not the shape. 

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.