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Essay Lab | Math Shack | Videos

Common Core Standards: Math

Grade 8

Geometry 8.G.A.4

4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Ever had a favorite piece of clothing made of wool? A cashmere scarf or an angora sock, perhaps? Or maybe you were the victim of your grandmother's itchy holiday sweater, which "accidentally" wound up in the dryer and shrunk to a third its normal size. Your grandma might not have been happy, but it was well worth it. You never got a wool sweater from her again and Mrs. Whiskerson looks more stylish than ever.

Similarity works the same way, only without the dryer sheets. Shapes are called similar when they have the same shape but different sizes. Even though the sweater shrunk, its measurements remained in proportion: if the sleeve was double the neckline, it stayed that way in its miniature form too.

Students should understand that two polygons are similar if they fulfill the following criteria:

  1. The polygons are the same shape. All corresponding angles in the polygons must be congruent.
  2. The sizes of the two polygons need not be the same. One polygon could be the size of your palm and the other could be the size of Antarctica.
  3. The corresponding sides of the polygons must be in proportion to each other. So when comparing corresponding sides, they should all yield the same ratio.

Students should already know that to verify that two polygons are congruent, all we need to do is prove that one can be carried onto the other using translation, reflection, and rotation only. Well, similarity is just the same way with an added bonus of dilation, free of charge.

Dilation is the only transformation that stands in between similarity and congruence because it preserves shape but changes size. Once two similar shapes have been dilated so that they're the same size, they should be congruent and can be moved using the congruence transformations.

In the end, this means that if two shapes can be carried onto one another using nothing but translation, rotation, reflection, and dilation, then they are similar to one another. Make sure your students remember that because it's pretty important. Yes, more important than shrinking more wool sweaters to fit Mrs. Whiskerson. Cats get itchy too, you know.

Drills

  1. Which of the following transformations will change the preimage into the image?

    Correct Answer:

    Dilation

    Answer Explanation:

    The two triangles are the same exact shape and they're oriented in the same exact way. All that needs to change is their size and luckily, that's what dilations are all about. Reflection and rotation (no matter how many degrees) will only change a figure's position or orientation, but not its size. Only dilation can take care of that.


  2. Which of the following is similar to the following polygon?

    Correct Answer:

    Answer Explanation:

    In order to be similar, two shapes can be different sizes but they must have the same exact angles. It's clear from the answer choices that the only shape with angles remotely matching those of the given figure is (D), which is a parallelogram. The only thing the other answer options have in common with the original parallelogram is that they're all quadrilaterals. Close, but no banana.


  3. Which of the following is true?

    Correct Answer:

    All squares are similar

    Answer Explanation:

    The two main aspects of similarity are same shape (also known as congruent angles) and different size (also known as proportional side lengths). Right triangles share one angle, but the other two can change willy-nilly. All the angles in a trapezoid can be different, so they're out too. Rectangles and squares have 90° all around, so they're safe for now. When it comes to proportional side lengths, the two dimensions of rectangles make them too loosey-goosey. After all a 1 × 2 rectangle isn't proportional to a 5 × 6 one. That leaves squares as our answer because their side lengths will always be proportional.


  4. Which of the following transformations would change a triangle into one that is similar, but not congruent?

    Correct Answer:

    Rotation and dilation

    Answer Explanation:

    If two triangles are similar and not congruent, they must be the same shape but not the same size. All the transformations mentioned except one are rigid motions, transformations that preserve size as well as shape. The only transformation listed that will shrink or enlarge a figure is dilation, so (D) is the right answer. All the others would produce a triangle that is congruent to the original one.


  5. Which of the following is true?

    Correct Answer:

    If two figures are congruent, then they are also similar

    Answer Explanation:

    What are the criteria for similar shapes again? They have to have identical corresponding angles, their side lengths must be in proportion, and they don't need to be the same size—but it doesn't say that they can't be. So two figures that are congruent are actually similar because they have congruent corresponding angles and side lengths that are proportional. The only specification we have about size is that it doesn't matter. (Hear that, ladies?) So congruent (read: identical) figures are also similar (read: the same shape, regardless of size).


  6. Which of the following is true about two squares of different sizes ABCD and WXYZ?

    Correct Answer:

    ABCD can be carried onto WXYZ using only dilation

    Answer Explanation:

    Translations and rotations move a square from one spot to another, but they aren't enough to change the size of the square. For that, we'd need dilation. You might think that (D) is a legit answer but it doesn't hold up once you know that all squares are similar to each other. (Consider their angles and side lengths!) That's why the right answer is (B).


  7. ΔABC can be changed into ΔDEF by a refection across an axis followed by a dilation. What would happen if the order of the transformations were changed?

    Correct Answer:

    This would yield the same result

    Answer Explanation:

    In this case, the order in which you do transformations doesn't matter. The dilation will always change the size and the reflection will always change the position and orientation. Whether we do 10 jumping jacks and then take 3 steps forward or take 3 steps forward and then do 10 jumping jacks doesn't matter because we'll end up in the same position (and equally tired).


  8. Which of the following is the best example of two similar objects that are not congruent?

    Correct Answer:

    A scale model of city hall and the actual city hall

    Answer Explanation:

    If two objects are congruent, they're the same in every single way—same size and shape. To be similar, they need to be exactly the same shape, but not the same size. The two cars in (A) are congruent, and the objects in (B) or (D) definitely don't have the same shape, so the only answer is (C). We're assuming that you don't live in Buckingham Palace. We apologize if we were wrong, Your Highness.


  9. Which of the following transformations would change two congruent figures into two similar (but not congruent) figures?

    Correct Answer:

    Translation and dilation

    Answer Explanation:

    Changing two congruent figures into two similar figures would require something other than just congruence transformations, so (C) is out. We need dilation to make similar figures, so (B) is out as well. The difference between (A) and (D) is that translation conserves shape, while a vertical stretch doesn't! For similarity, we need the figures to stay the same shape, so (A) is our only option.


  10. Which of the following is false?

    Correct Answer:

    Reflection is a congruence transformation only

    Answer Explanation:

    Congruence transformations conserve both the size and shape of a figure, but you already knew that. Translation, reflection, and rotation are congruence transformations, so we know that (D) is true. Dilation conserves only shape because (B) is true, so we're left with (A) and (C). Since similarity transformations have to conserve a figure's shape, all congruence transformations are similarity transformations, too. That means (A) is true and (C) is false, which is the right answer.