pAIrmatic Lessons

Similarity Through Dilations

8.G.4Grade 8Geometry20 practice questions

Two figures are similar when one is a scaled copy of the other: same shape, maybe a different size. A dilation does the scaling; slides, flips and turns put the copy in place.

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The big idea

A dilation changes size but keeps shape: every angle stays the same, and every length is multiplied by the scale factor k. Rigid motions (translations, reflections and rotations) keep both size and shape.

So two figures are similar (~) if a dilation followed by rigid motions maps one onto the other. Similar figures have equal matching angles, and every pair of matching sides is in the same ratio.

To find a missing side, get the scale factor from one known pair (image ÷ original), then multiply. To test two figures, match shortest side with shortest and longest with longest, and check that every pair gives the same ratio.

Words you need

Similar (~)
Same shape, possibly a different size: a dilation and rigid motions map one onto the other.
Scale factor
The number every length is multiplied by. Above 1 enlarges, between 0 and 1 shrinks.
Corresponding sides
Sides that land on each other when one figure is mapped onto the other.
Proportional
In the same ratio: 4 to 10 and 6 to 15 both simplify to 2 to 5.

Worked examples

Example 1

Triangle ABC has sides 4, 6 and 8. It is similar to triangle DEF, whose matching sides are 10, 15 and an unknown side. Find the unknown side.

  1. Scale factor = image ÷ original = 10 ÷ 4 = 2.5.
  2. Check with a second pair: 15 ÷ 6 = 2.5. Same ratio, so the sides match up.
  3. Unknown side = 8 × 2.5 = 20.

Answer: 20

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Example 2

Triangle A has vertices (1, 1), (3, 1) and (1, 2). Triangle B has vertices (5, 1), (9, 1) and (5, 3). Find a dilation and a translation that map A onto B.

  1. Compare one pair of sides: the bottom side grows from 2 to 4, so the scale factor is 2.
  2. Dilate A by 2 about the origin: (2, 2), (6, 2), (2, 4).
  3. Each of those points needs to move 3 right and 1 down to reach B: (5, 1), (9, 1), (5, 3). Every vertex matches.

Answer: (x, y) → (2x, 2y) then (x, y) → (x + 3, y − 1)

−1−11122334455667788991010ABCA′B′C′

Example 3

Is a triangle with sides 5, 7 and 9 similar to a triangle with sides 10, 14 and 20?

  1. Match the sides shortest to longest: 5 with 10, 7 with 14, 9 with 20.
  2. Ratios: 10 ÷ 5 = 2, 14 ÷ 7 = 2, 20 ÷ 9 ≈ 2.2.
  3. The ratios aren't all equal, so the triangles are not similar.

Answer: No, not similar

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Common mistakes

Flipping the ratio partway through: image ÷ original for one pair, original ÷ image for the next. Pick one order, such as image ÷ original, and use it for every pair.

Adding instead of multiplying: a side that grows from 4 to 6 doesn't mean every side grows by 2. Similar figures grow by the same factor, not the same amount. 4 to 6 is × 1.5, so a 10 side becomes 15.

Matching sides by where they sit on the page. A flipped or turned copy lists its sides in a new order. Match shortest to shortest and longest to longest.

Practice: 20 questions

12 of 20 are word problems. For a word problem, sketch it first and mark what you know.

Tier 1 Foundations Use the rule directly

  1. Triangle ABC ~ triangle DEF. The sides of ABC are 3, 5 and 6. The matching sides of DEF are 6, 10 and an unknown side. Find it.
    356610?Not drawn to scale
  2. A triangle with sides 3, 4 and 6 is similar to a triangle with sides 9, 12 and 18. What is the scale factor from the small triangle to the large one?
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  3. A rectangular flag is 3 feet tall and 5 feet long. A similar flag is 9 feet tall. How long is it? Word problem
  4. A logo is 8 cm wide and 5 cm tall. A similar copy on a banner is 40 cm wide. How tall is the copy? Word problem
  5. Kofi draws a triangle whose shortest side is 5 cm and longest side is 8 cm. He draws a similar triangle with a shortest side of 15 cm. How long is its longest side? Word problem

Tier 2 Intermediate One more step to think through

  1. Which dilation about the origin maps triangle A with vertices (2, 2), (4, 2), (2, 6) onto triangle B with vertices (1, 1), (2, 1), (1, 3)?
    −1−111223344556677ABCA′B′C′
  2. Is a triangle with sides 4, 6 and 7 similar to a triangle with sides 9, 10.5 and 6?
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  3. A postcard is 6 inches tall. A similar poster is 18 inches tall and 27 inches wide. How wide is the postcard? Word problem
  4. Ayla says a 2 by 3 rectangle is similar to a 4 by 5 rectangle, because each side grew by 2. Is she right? Word problem
  5. A designer shrinks a triangle with corners (0, 0), (8, 0) and (0, 6) about the origin by a scale factor of 1/2. Where do the corners land? Word problem
  6. A rectangle 5 cm by 8 cm is dilated to a rectangle 12.5 cm by 20 cm. What scale factor was used? Word problem
  7. Find a dilation followed by a translation that maps triangle ABC with A(1, 0), B(2, 0), C(1, 2) onto triangle DEF with D(3, 1), E(6, 1), F(3, 7).
    −1−11122334455667788ABCA′B′C′

Tier 3 Advanced Several steps - find the hidden structure

  1. Triangle P has sides 6, 9 and 12. Triangle Q is similar to P and has a perimeter of 54. How long is the shortest side of Q? Word problem
  2. Find a reflection and a dilation that map triangle A with vertices (2, 1), (4, 1), (2, 2) onto triangle B with vertices (−4, 2), (−8, 2), (−4, 4).
    −9−9−7−7−5−5−3−3−1−1113355ABCA′B′C′
  3. One kite frame is a triangle with corners (0, 0), (4, 0) and (0, 3). Another has corners (2, 1), (10, 1) and (2, 7). Are the frames similar? If so, what is the scale factor? Word problem
  4. An 8 cm by 10 cm photo gets a border that makes the framed picture 12 cm by 14 cm. Is the framed picture similar to the photo? Word problem
  5. A copier enlarges a drawing by a scale factor of 1.5, then enlarges the copy by 1.5 again. A line is 8 cm long on the original. How long is it on the final copy? Word problem

Tier 4 Challenge Stretch problems, some above grade level

  1. Dilate the triangle with vertices (2, 1), (4, 1), (2, 3) by a scale factor of 2 about the point (2, 1). Above grade level
    −1−11122334455ABCorange dot = center (2, 1)
  2. Two triangular sails are similar. The small one has an area of 12 square feet, and the large one's sides are 2.5 times as long. What is the area of the large sail? Word problem
  3. Find a rotation and a dilation about the origin that map the triangle with vertices (0, 0), (4, 2), (2, 6) onto the triangle with vertices (0, 0), (−3, 6), (−9, 3).
    −10−10−8−8−6−6−4−4−2−2224466ABCA′B′C′

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