Which option is a scaled version of image P?

Question

You have to design a water bottle for a Kindergarten child. Which factors will you keep in mind?

Answer 1

Step 1: Recall the diagonal test.
Designers check whether one rectangle is a scaled copy of another by placing them corner to corner and extending a single diagonal line from the shared corner through the far corner of the larger one.

Step 2: Apply it here.
Line up image P's bottom-left corner with each option's bottom-left corner, then draw the diagonal from that shared corner through P's own top-right corner.

Step 3: Check where each option's far corner falls.
For a true scaled copy, its own top-right corner must land exactly on this diagonal line, since scaling moves a corner only along the line joining it to the shared corner.

Step 4: Identify the match.
Only image C's top-right corner falls on this diagonal line drawn from P; A, B and D each fall off the line, in the direction of whichever way they have been stretched.
\[ \boxed{\text{C}} \]

Answer 2

A third way to check is with the angle a rectangle's diagonal makes with its base, since that angle is fixed by the width-to-height ratio alone, \( \theta = \arctan\left(\frac{\text{height}}{\text{width}}\right) \), and stays the same for any true scaled copy.

A: The diagonal here sits at a steeper angle than in P, since the shape has grown taller relative to its width, so \( \theta \) does not match P.
B: The diagonal here sits at a shallower angle than in P, since the shape has grown wider relative to its height, so \( \theta \) again does not match.
C: The diagonal here sits at the same angle as in P, just over a shorter diagonal length, since the whole shape has simply been shrunk.
D: The diagonal here sits at a shallower angle than in P as well, since the shape has been squashed vertically, so \( \theta \) does not match.

Only C keeps the same diagonal angle \( \theta \) as P.

So the correct answer is C.

Answer 3

A third way to check is with the angle a rectangle's diagonal makes with its base, since that angle is fixed by the width-to-height ratio alone, \( \theta = \arctan\left(\frac{\text{height}}{\text{width}}\right) \), and stays the same for any true scaled copy.

A: The diagonal here sits at a steeper angle than in P, since the shape has grown taller relative to its width, so \( \theta \) does not match P.
B: The diagonal here sits at a shallower angle than in P, since the shape has grown wider relative to its height, so \( \theta \) again does not match.
C: The diagonal here sits at the same angle as in P, just over a shorter diagonal length, since the whole shape has simply been shrunk.
D: The diagonal here sits at a shallower angle than in P as well, since the shape has been squashed vertically, so \( \theta \) does not match.

Only C keeps the same diagonal angle \( \theta \) as P.

So the correct answer is C.