Answer:
Part 1) The perimeter of the new figure must be equal to the perimeter of the original figure multiplied by the scale factor (see the explanation)
Part 2) The area of the new figure must be equal to the area of the original figure multiplied by the scale factor squared
Part 3) The new figure and the original figure are not similar figures (see the explanation)
Step-by-step explanation:
Part 1) If a scale factor is applied to a figure and all dimensions are changed proportionally, what is the effect on the perimeter of the figure?
we know that
If all dimensions are changed proportionally, then the new figure and the original figure are similar
When two figures are similar, the ratio of its perimeters is equal to the scale factor
so
The perimeter of the new figure must be equal to the perimeter of the original figure multiplied by the scale factor
Part 2) If a scale factor is applied to a figure and all dimensions are changed proportionally, what is the effect on the area of the figure?
we know that
If all dimensions are changed proportionally, then the new figure and the original figure are similar
When two figures are similar, the ratio of its areas is equal to the scale factor squared
so
The area of the new figure must be equal to the area of the original figure multiplied by the scale factor squared
Part 3) What would happen to the perimeter and area of a figure if the dimensions were changed NON-proportionally? For example, if the length of a rectangle was tripled, but the width did not change? Or if the length was tripled and the width was decreased by a factor of 1/4?
we know that
If the dimensions were changed NON-proportionally, then the ratio of the corresponding sides of the new figure and the original figure are not proportional
That means
The new figure and the original figure are not similar figures
therefore
Corresponding sides are not proportional and corresponding angles are not congruent
so
<em>A) If the length of a rectangle was tripled, but the width did not change?</em>
<em>Perimeter</em>
The original perimeter is P=2L+2W
The new perimeter would be P=2(3L)+2W ----> P=6L+2W
The perimeter of the new figure is greater than the perimeter of the original figure but are not proportionals
<u>Area</u>
The original area is A=LW
The new area would be A=(3L)(W) ----> A=3LW
The area of the new figure is three times the area of the original figure but its ratio is not equal to the scale factor squared, because there is no single scale factor
<em>B) If the length was tripled and the width was decreased by a factor of 1/4?</em>
<u>Perimeter</u>
The original perimeter is P=2L+2W
The new perimeter would be P=2(3L)+2(W/4) ----> P=6L+W/2
The perimeter of the new figure and the perimeter of the original figure are not proportionals
<u><em>Area</em></u>
The original area is A=LW
The new area would be A=(3L)(W/4) ----> A=(3/4)LW
The area of the new figure is three-fourth times the area of the original figure but its ratio is not equal to the scale factor squared, because there is no single scale factor
