Similarity

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1. Dilations

Warm-up

Three quick questions from earlier lessons.

Discover

A photo is enlarged so every length doubles. What happens to the angles in the photo?

  1. They stay the same
  2. They double

Change the scale factor k. What happens to the angle at A′?

  1. It stays 90°
  2. It is multiplied by k too

Key idea

A dilation stretches from a center.

A dilation with center OO and scale factor k>0k > 0 sends each point PP to P′P' on ray OPOP with OP′=k⋅OPOP' = k \cdot OP. About the origin: (x,y)→(kx,ky)(x, y) \to (kx, ky). Lengths are multiplied by kk; angles are unchanged.

Worked example

Dilate A(2,−1)A(2, -1) about the origin with k=3k = 3.

  1. Step 1Triangle ABC and the center O.
  2. Step 2Each point slides along its ray from O.
  3. Step 3k = 2: every distance from O doubles; the angles do not change.

A′=(6,−3)A' = (6, -3)
Multiply both coordinates.
OA′=3⋅OAOA' = 3 \cdot OA
Check the distance.

Your turn, guided

Your turn: a segment of 4 becomes 10. The scale factor.

2.52.5
10 ÷ 4.

Practice ★

  1. Dilate A(3,1)A(3, 1) about the origin with scale factor 2. Tap its image.
  2. Dilate A(−3,1)A(-3, 1) about the origin with scale factor 3. Tap its image.
  3. Dilate A(−3,−1)A(-3, -1) about the origin with scale factor 2. Tap its image.

Watch out!

Practice ★★

  1. A dilation takes a segment of length 3 to a segment of length 11. Find the scale factor.
  2. A dilation takes a segment of length 3 to a segment of length 26. Find the scale factor.
  1. A triangle is dilated with scale factor 5. What happens to its angles?
    1. They stay the same
    2. They are multiplied by 5
    3. They grow by 5 degrees

Exit ticket

Three questions to check you have got it.

  1. Dilate A(3,1)A(3, 1) about the origin with scale factor 2. Tap its image.
  2. Dilate A(−3,1)A(-3, 1) about the origin with scale factor 3. Tap its image.
  3. Dilate A(−3,−1)A(-3, -1) about the origin with scale factor 2. Tap its image.

Summary card

Dilations

(x,y)→(kx,ky)(x, y) \to (kx, ky) about the origin; lengths × kk, angles kept.

Watch out: multiply, do not add.

2. Similar figures and scale factors

Warm-up

Three quick questions from earlier lessons.

Discover

A 2 by 3 rectangle is enlarged with scale factor 2. What is the area of the new rectangle?

  1. 24
  2. 12

Key idea

Similar figures: a dilation (and rigid moves) apart.

Two figures are similar if a sequence of rigid transformations and dilations maps one onto the other: corresponding angles are equal and corresponding lengths are in the same ratio kk. Perimeters scale by kk, areas by k2k^2 (and volumes by k3k^3).

Worked example

Similar triangles with scale factor 3; the small one has perimeter 12 and area 6.

Perimeter 3636
Lengths × 3.
Area 5454
Areas × 9.

Your turn, guided

Your turn: scale factor 4, small area 5. The large area.

8080
5 × 16.

Practice ★

  1. Two polygons are similar. A side of length 5 corresponds to a side of length 15. Another side of the small polygon has length 9. Find the matching side of the large one.
  2. Two polygons are similar. A side of length 5 corresponds to a side of length 20. Another side of the small polygon has length 4. Find the matching side of the large one.
  1. A polygon has perimeter 20. It is enlarged by a scale factor of 5. Find the new perimeter.

Watch out!

Practice ★★

  1. A shape has area 9. It is enlarged by a scale factor of 2. Find the new area.
  2. A shape has area 22. It is enlarged by a scale factor of 4. Find the new area.

Exit ticket

Three questions to check you have got it.

  1. Two polygons are similar. A side of length 5 corresponds to a side of length 15. Another side of the small polygon has length 9. Find the matching side of the large one.
  2. Two polygons are similar. A side of length 5 corresponds to a side of length 20. Another side of the small polygon has length 4. Find the matching side of the large one.
  3. Two polygons are similar. A side of length 3 corresponds to a side of length 15. Another side of the small polygon has length 8. Find the matching side of the large one.

Summary card

Similar figures

Lengths × kk, perimeters × kk, areas × k2k^2.

Watch out: scale by multiplying, and square kk for areas.

3. Triangle similarity

Warm-up

Three quick questions from earlier lessons.

Discover

Two triangles each have angles 40∘40^\circ and 60∘60^\circ. What is the third angle in each?

  1. 80∘80^\circ in both
  2. It depends on the sizes

Key idea

Two angles are enough.

AA: two pairs of equal angles make triangles similar. (Also SSS similarity: all sides proportional; SAS similarity: two sides proportional and the included angles equal.) Then corresponding sides are proportional: DEAB=EFBC=DFAC\dfrac{DE}{AB} = \dfrac{EF}{BC} = \dfrac{DF}{AC}.

Worked example

△ABC∼△DEF\triangle ABC \sim \triangle DEF, AB=4AB = 4, BC=6BC = 6, DE=10DE = 10.

Scale factor 2.52.5
DE/AB.
EF=15EF = 15
EF = 2.5 × BC.

Your turn, guided

Your turn: AB=3AB = 3, AC=5AC = 5, DE=9DE = 9. Find DFDF.

DF=15DF = 15
5 × 3.

Practice ★

  1. Two triangles each have a 76∘76^\circ angle and a 50∘50^\circ angle. What can you conclude?
    1. They are similar (AA)
    2. They are congruent
    3. Nothing without side lengths
  1. △ABC∼△DEF\triangle ABC \sim \triangle DEF with AB=8AB = 8, BC=7BC = 7 and DE=25DE = 25. Find EFEF.Not to scale
  2. △ABC∼△DEF\triangle ABC \sim \triangle DEF with AB=9AB = 9, BC=3BC = 3 and DE=24DE = 24. Find EFEF.Not to scale

Watch out!

Practice ★★

  1. Which figures are always similar to each other?
    1. Any two squares (for example sides 7 and 14)
    2. Any two rectangles
    3. Any two isosceles triangles

Exit ticket

Three questions to check you have got it.

  1. △ABC∼△DEF\triangle ABC \sim \triangle DEF with AB=8AB = 8, BC=7BC = 7 and DE=25DE = 25. Find EFEF.Not to scale
  2. △ABC∼△DEF\triangle ABC \sim \triangle DEF with AB=9AB = 9, BC=3BC = 3 and DE=24DE = 24. Find EFEF.Not to scale
  3. △ABC∼△DEF\triangle ABC \sim \triangle DEF with AB=7AB = 7, BC=9BC = 9 and DE=16DE = 16. Find EFEF.Not to scale

Summary card

Triangle similarity

AA (or SSS, SAS similarity) → corresponding sides proportional.

Watch out: AA works for triangles only.

4. Proportional segments: the side-splitter theorem

Warm-up

Three quick questions from earlier lessons.

Discover

A line parallel to one side of a triangle cuts the other two sides. If it cuts one side in the ratio 1 : 2, how does it cut the other?

  1. Also 1:21 : 2
  2. It depends on the angle

Slide DE. Divide AD by DB, and AE by EC.

  1. The two quotients are always equal
  2. AD always equals AE

Key idea

Parallel lines split sides proportionally.

Side-splitter theorem: if DE∥BCDE \parallel BC with DD on ABAB and EE on ACAC, then ADDB=AEEC\dfrac{AD}{DB} = \dfrac{AE}{EC}, and △ADE∼△ABC\triangle ADE \sim \triangle ABC. The segment joining two midpoints (a midsegment) is parallel to the third side and half as long.

Worked example

AD=4AD = 4, DB=6DB = 6, AE=6AE = 6, DE∥BCDE \parallel BC.

46=6EC\dfrac{4}{6} = \dfrac{6}{EC}
AD/DB = AE/EC.
EC=9EC = 9
Cross-multiply.

Your turn, guided

Your turn: BC=18BC = 18. The midsegment parallel to BCBC.

99
Half.

Practice ★

  1. In △ABC\triangle ABC, a line parallel to BCBC meets ABAB at DD and ACAC at EE, with AD=5AD = 5, DB=3DB = 3 and AE=9AE = 9. Find ECEC.Not to scale
  2. In △ABC\triangle ABC, a line parallel to BCBC meets ABAB at DD and ACAC at EE, with AD=4AD = 4, DB=3DB = 3 and AE=9AE = 9. Find ECEC.Not to scale
  1. In a triangle, BC=26BC = 26. The segment joining the midpoints of ABAB and ACAC has length…

Watch out!

Practice ★★

  1. DE∥BCDE \parallel BC with DD on ABAB, EE on ACAC; AD=2AD = 2, DB=7DB = 7, AE=3AE = 3. Find ACAC.Not to scale
  1. Boss problem. In △ABC\triangle ABC, DE∥BCDE \parallel BC with DD on ABAB, AD=6AD = 6, DB=1DB = 1 and DE=8DE = 8.

Exit ticket

Three questions to check you have got it.

  1. In △ABC\triangle ABC, a line parallel to BCBC meets ABAB at DD and ACAC at EE, with AD=5AD = 5, DB=3DB = 3 and AE=9AE = 9. Find ECEC.Not to scale
  2. In △ABC\triangle ABC, a line parallel to BCBC meets ABAB at DD and ACAC at EE, with AD=4AD = 4, DB=3DB = 3 and AE=9AE = 9. Find ECEC.Not to scale
  3. In △ABC\triangle ABC, a line parallel to BCBC meets ABAB at DD and ACAC at EE, with AD=7AD = 7, DB=8DB = 8 and AE=5AE = 5. Find ECEC.Not to scale

Summary card

Proportional segments

DE∥BCDE \parallel BC: AD/DB=AE/ECAD/DB = AE/EC; midsegment = half the third side.

Watch out: keep the parts in matching order.

5. Similar right triangles and the Pythagorean theorem

Warm-up

Three quick questions from earlier lessons.

Discover

The altitude to the hypotenuse of a right triangle creates two smaller right triangles. How are they related to the original?

  1. All three triangles are similar
  2. They are congruent to each other

Key idea

Similar right triangles give the Pythagorean theorem.

If the altitude hh cuts the hypotenuse cc into pp and qq: h2=pqh^2 = pq; each leg satisfies a2=pca^2 = pc (with pp next to aa), b2=qcb^2 = qc. Adding: a2+b2=c(p+q)=c2a^2 + b^2 = c(p + q) = c^2.

Worked example

Segments p=4p = 4, q=9q = 9.

h=36=6h = \sqrt{36} = 6
h² = pq.
a=52a = \sqrt{52}
c = 13; a² = 4 × 13.
b=117b = \sqrt{117}
b² = 9 × 13.

Your turn, guided

Your turn: legs 6 and 8. The hypotenuse.

1010
√(36 + 64).

Practice ★

  1. A right triangle has legs 9 and 12. Find the hypotenuse.
  2. A right triangle has legs 3 and 4. Find the hypotenuse.
  1. The altitude to the hypotenuse of a right triangle cuts the hypotenuse into segments 36 and 49. Find the altitude.
  2. The altitude to the hypotenuse of a right triangle cuts the hypotenuse into segments 9 and 16. Find the altitude.

Watch out!

Practice ★★

  1. In a right triangle, the hypotenuse is 25 and the altitude cuts off a segment of 1 next to leg aa. Find aa.
  2. In a right triangle, the hypotenuse is 49 and the altitude cuts off a segment of 1 next to leg aa. Find aa.

Exit ticket

Three questions to check you have got it.

  1. A right triangle has legs 9 and 12. Find the hypotenuse.
  2. A right triangle has legs 3 and 4. Find the hypotenuse.
  3. A right triangle has legs 18 and 24. Find the hypotenuse.

Summary card

Similar right triangles

h2=pqh^2 = pq, a2=pca^2 = pc, b2=qcb^2 = qc; hence a2+b2=c2a^2 + b^2 = c^2.

Watch out: geometric mean, not average.

Answers: Similarity

1. Dilations

A

  1. (6,2)(6, 2)
  2. (−9,3)(-9, 3)
  3. (−6,−2)(-6, -2)

B

  1. 113\frac{11}{3}
  2. 263\frac{26}{3}

C

  1. They stay the same

D

  1. (6,2)(6, 2)
  2. (−9,3)(-9, 3)
  3. (−6,−2)(-6, -2)

2. Similar figures and scale factors

A

  1. 2727
  2. 1616

B

  1. 100100

C

  1. 3636
  2. 352352

D

  1. 2727
  2. 1616
  3. 4040

3. Triangle similarity

A

  1. They are similar (AA)

B

  1. 1758\frac{175}{8}
  2. 88

C

  1. Any two squares (for example sides 7 and 14)

D

  1. 1758\frac{175}{8}
  2. 88
  3. 1447\frac{144}{7}

4. Proportional segments: the side-splitter theorem

A

  1. 275\frac{27}{5}
  2. 274\frac{27}{4}

B

  1. 1313

C

  1. 272\frac{27}{2}

D

  1. 4936\frac{49}{36}

E

  1. 275\frac{27}{5}
  2. 274\frac{27}{4}
  3. 407\frac{40}{7}

5. Similar right triangles and the Pythagorean theorem

A

  1. 1515
  2. 55

B

  1. 4242
  2. 1212

C

  1. 55
  2. 77

D

  1. 1515
  2. 55
  3. 3030