Suppose you find the ratio of the lengths of adjacent sides in a parallelogram. This ratio is equivalent to the ratio of the adjacent sides in another parallelogram. Are the figures similar? Explain.
Isn't this a question in a 7th grade savaas book?
Yes, the figures are similar if the ratio of the lengths of adjacent sides in one parallelogram is equivalent to the ratio of the adjacent sides in another parallelogram.
To understand why, we need to look at the definition of similar figures. Two figures are considered similar if their corresponding angles are congruent, and the ratios of the lengths of their corresponding sides are equal.
In the given scenario, we have two parallelograms. Let's call the lengths of the adjacent sides in the first parallelogram "a" and "b," and the lengths of the adjacent sides in the second parallelogram "c" and "d."
If the ratio of a:b in the first parallelogram is equivalent to the ratio of c:d in the second parallelogram, then we have:
a/b = c/d
To prove that the figures are similar, we need to show that the corresponding angles are congruent. In a parallelogram, opposite angles are congruent, so we have:
∠1 = ∠3 and ∠2 = ∠4
Additionally, we can use the opposite sides of the parallelogram property. In a parallelogram, the opposite sides are congruent, so we have:
Segment AB = Segment CD and Segment BC = Segment DA
Now, let's analyze the corresponding sides:
In the first parallelogram, the adjacent sides have a ratio of a:b. For example, Segment AB : Segment BC has a ratio of a:b.
In the second parallelogram, the adjacent sides have a ratio of c:d. For example, Segment CD : Segment DA has a ratio of c:d.
Since a/b = c/d, the ratio of the corresponding sides is equal. Therefore, the figures have congruent corresponding angles and equal ratios of corresponding side lengths, fulfilling the conditions for similarity.
Thus, we can conclude that the figures are similar.
To determine whether the figures are similar, we need to compare their corresponding angles and side lengths.
Parallel lines in a parallelogram form pairs of congruent angles, which means that the corresponding angles in the two parallelograms are congruent. Therefore, the corresponding angles are equal.
Now, let's focus on the ratio of the lengths of adjacent sides. Suppose the ratio of the lengths of adjacent sides in the first parallelogram is a:b. We can represent this as a/b.
According to the given information, the ratio of the lengths of the adjacent sides in the second parallelogram is also a:b. Therefore, both parallelograms have sides with equal ratios, indicating that their corresponding side lengths are proportional.
When corresponding angles are congruent and corresponding side lengths are proportional, the figures are similar. In this case, the first and second parallelograms are similar.
Not necessarily.
The angles could be different, thus not similar
Imagine the second figure having hinges at the vertices and you "squish" the figure.