3. Which of these situations is not possible if two shapes are similar? Show your work & explain!
A The perimeter of one is triple the perimeter of another, and the side
lengths are also triple.
B The perimeter of one is twice the perimeter of another, and the area
is four times as much.
C The angles of one and the angles of the other are the same, but
one area is three times as much as the other.
D The perimeter of one is triple the perimeter of another, and the area
is also triple.
A situation where the perimeter of one shape is triple the perimeter of another, and the side lengths are also triple is possible if two shapes are similar. This is because if two shapes are similar, their corresponding sides are in proportion, meaning that the ratio of corresponding side lengths is the same. If the side lengths are tripled, then the perimeters will also be tripled.
B A situation where the perimeter of one shape is twice the perimeter of another, and the area is four times as much is also possible if two shapes are similar. This is because the ratio of corresponding side lengths is the same for similar shapes, and if the side lengths are doubled, the perimeter will also be doubled. However, the ratio of corresponding areas for similar shapes is the square of the ratio of corresponding side lengths. So if the side lengths are doubled, the area will be four times as much.
C A situation where the angles of one shape are the same as the angles of another, but one area is three times as much as the other is not possible if two shapes are similar. This is because if two shapes are similar, their corresponding angles are congruent. However, the ratio of corresponding areas for similar shapes is the square of the ratio of corresponding side lengths. So if the side lengths are tripled, the area will be nine times as much, not three times as much. Therefore, this situation is not possible if two shapes are similar.
D A situation where the perimeter of one shape is triple the perimeter of another, and the area is also triple is possible if two shapes are similar. This is because if two shapes are similar, their corresponding side lengths are in proportion, meaning that the ratio of corresponding side lengths is the same. If the side lengths are tripled, then the perimeters will also be tripled. Additionally, the ratio of corresponding areas for similar shapes is the square of the ratio of corresponding side lengths. So if the side lengths are tripled, the area will be nine times as much, which is also triple. Therefore, this situation is possible if two shapes are similar.