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Common Core Standards: Math

High School: Geometry

Similarity, Right Triangles, and Trigonometry HSG-SRT.A.1

1. Verify experimentally the properties of dilations given by a center and a scale factor:

Your students may understand generally what the term similar means, but it can be very specific when it comes to geometry. For instance, while Tia and Tamera Mowry are very similar (you know, since they're identical twins and all), they aren't geometrically similar. How on earth could that be?

In geometry, similar objects are exactly the same shape, but not necessarily the same size. So one object could be smaller than a pea and another could be larger than Antarctica, but if they have the exact same shape, they're similar. How curious.

Still, the idea of being "exactly the same shape" is a little vague. How can we be sure that two objects of different sizes still have the same shape? That's where dilations, centers, and scale factors come in.

Standard Components

Drills

  1. What is a scale factor?

    Correct Answer:

    The number by which the distance from the center of dilation to an object is multiplied by to obtain a similar object as measured from the center of dilation to the dilated object

    Answer Explanation:

    The scale factor is the number by which an object is multiplied by to create a similar object. The only reason (B) is wrong is because it has us subtract the scale factor rather than multiply it. The center of dilation and the scale factor are related, but different. While (D) is related to the scale factor too, it's a ratio of the distances rather than the distance between them.


  2. Using the white triangle as the original image, which of the following uses the right angle vertex as the center to dilate the triangle by a scale factor of 2?

    Correct Answer:

    Answer Explanation:

    Our right angle vertex is the center, so let's pick another vertex to use as a reference. Start by drawing lines from that point to each corresponding point on both triangles. A is the only choice where you can easily draw straight lines from the right angle to each vertex and the lines go through both triangles. (Don't be confused by the fact that the white triangle is on top of the blue triangle.) Drawing lines in answers (B) and (C) will just create a lot of mess.


  3. If the blue figure is an object and the red figure is the object after the dilation, what is the scale factor?

    Correct Answer:

    0.5

    Answer Explanation:

    The easiest way to do this is to take a vertex from the blue triangle, like (6, 4). The corresponding vertex in the red triangle is (3, 2). To go from 6 to 3 and 4 to 2, you divide each by 2 (the scale factor). Voilà! To check, we can do the same thing with another vertex: (2, 6) goes to (1, 3). Again, that means we divide by 2. Since the final object is smaller than the original image, our scale factor has to be 0.5, not 2.


  4. Which of the following statements is true about the figure?

    Correct Answer:

    ΔA'B'C' has been dilated using O as the center.

    Answer Explanation:

    Even though this section isn't about congruence, (A) is not right because the triangles are different sizes. If ΔABC creates ΔA'B'C', then it would have to have a scale factor of less than 1 (to make the image smaller). Also, how can we dilate an image using a center that we can only find after dilating the image? The only answer that makes sense is (C).


  5. Which point is the center of dilation for the following object (ΔABC) and its dilated object (ΔA'B'C')?

    Correct Answer:

    D

    Answer Explanation:

    For a point to be the center of dilation, we should be able to draw straight lines that intersect with both A and A', B and B', and C and C'. If we imagine doing so, all three lines would intersect at D. We know that (A), (C), and (D) are wrong because there's no way one line could pass through E, C, and C' or through F, A, and A'. Same goes for the origin.


  6. Compared to ΔABC, which of the following is a similar object with center of dilation at D and a scale factor of 0.5?

    Correct Answer:

    Answer Explanation:

    Only (A) shows the correct dilation and scale factor. We know the original triangle has to be twice as big as the new triangle, and the new triangle has to be in between D and the original triangle. The only image that applies is (A). All the others are reflections or have nothing to do with D (and sometimes, very little to do with ΔABC).


  7. Which of the following correctly plots a triangle at (0, 0), (2, 3), and (3, 1), and then dilates it with a scale factor of one third, using the origin as the center of dilation?

    Correct Answer:

    Answer Explanation:

    We need two triangles, the original being 3 times larger than the dilated one. We can automatically eliminate (A) and (D), but we have to be careful when choosing between (B) and (C). Again, the original triangle, ΔABC, has to be the larger one. That means (C) is right.


  8. Assuming the center of dilation is the origin in the following picture, what would the scale factor be?

    Correct Answer:

    3

    Answer Explanation:

    The coordinates have been multiplied by 3 in each case from the original object to the dilated one. Even though (D) is the center of dilation, it's not the scale factor, and therefore wrong. A scale factor of 0 wouldn't even produce an object, and a scale factor of 1 would produce the exact same object. The only logical answer is (B).


  9. Using the AB as the original object and C as the center of dilation, which of the following is true?

    Correct Answer:

    The segment was dilated twice.

    Answer Explanation:

    The only correct answer is (D) because the segment was dilated twice to create A'B' and A''B''. A segment can be the original object (and indeed it was), but it wasn't dilated three times because there still has to be an original object. No scale factor means no dilation, so (B) is impossible, too.


  10. From the diagram, choose the correct scale factor.

    Correct Answer:

    2

    Answer Explanation:

    The only center of dilation that would allow one to draw a line through each vertex is D. Since the ratio of the side lengths is 2, the scale factor is 2 as well. (Remember that we look at the ratio, not the difference, to find the scale factor!) We also have to make sure to put the final lengths in the numerator and the initial lengths in the denominator, otherwise we may end up with ½ as the scale factor, which is incorrect.