Geometry path · G07

Before this lesson: Parallel lines and angle bisectors, Angle chasing

Your goal: Identify a correspondence and distinguish length scaling from area scaling.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Names you may know: triangle similarity; AA, SAS and SSS similarity.

Similar triangles: the key idea

Triangles are similar when their corresponding angles agree and side lengths have one common scale factor. AA, SAS with proportional enclosing sides, and SSS with all side ratios equal are standard tests. Write the vertices in corresponding order before using ratios. A scale k multiplies lengths and perimeters by k but areas by k². Parallel lines, equal circle angles and a shared angle can reveal similarity in overlapping triangles. Similarity alone does not imply equal size.

Triangle ABC with marked points on its sides.ABCDE
DE is parallel to BC. Triangles ADE and ABC have matching corresponding angles.

A worked example

Triangles ABC and DEF are similar in that order. AB=3,BC=4,CA=5 and DE=6. Find EF,FD and the area ratio.

AB corresponds to DE, so the scale from ABC to DEF is 6/3=2. Thus EF=8 and FD=10. The area of DEF is 2²=4 times that of ABC.

Your turn: change one thing

Two similar triangles have area ratio 9:25. What is the ratio of corresponding sides?

Try this on paper before opening the explanation.

Compare your reasoning

Lengths are positive, so take the positive square root: 3:5.

Pause and check

A trap to avoid: Using the length scale as the area scale.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Prove that areas of similar triangles are in the square of their side ratio.

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Take corresponding bases b,kb. Corresponding altitudes are also scaled by k, because the perpendicular right triangles have the same angles. Their areas are bh/2 and (kb)(kh)/2=k²bh/2. Hence the area ratio is k².

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. AA stands for what similarity condition?

Foundation

  1. Any two angles in one triangle agree
  2. Two corresponding angles agree
  3. All areas agree
  4. Two arbitrary sides agree
Hint

The third angle then agrees too.

Answer and reasoning

Two corresponding angles agree. The triangle angle sum determines the remaining angle.

2. If the length scale is k, the area scale is what?

Foundation

  1. 2k
  2. k\sqrt{k}
  3. k²
  4. k
Hint

Both base and height scale.

Answer and reasoning

k². Area is one half times base times height, so both contribute k.

3. A 3–4–5 triangle scaled by 2 has sides what?

Core

  1. 5,6,7
  2. 9,16,25
  3. 3,8,15
  4. 6,8,10
Hint

Multiply every side by the same factor.

Answer and reasoning

6,8,10. The scale must be consistent for all corresponding sides.

4. Area ratio 9:25 gives side ratio what?

Core

  1. 5:3 always
  2. 3:5
  3. 9:25
  4. 81:625
Hint

Take positive square roots in the same order.

Answer and reasoning

3:5. 9:25=3:5\sqrt{9}:\sqrt{25}=3:5.

5. Why write vertices in corresponding order?

Stretch

  1. Because labels determine lengths
  2. To form ratios between matching sides
  3. To make every triangle congruent
  4. To avoid angles
Hint

Mispaired sides lead to false ratios.

Answer and reasoning

To form ratios between matching sides. The order ABC∼DEF identifies A↔D,B↔E,C↔F.

6. Do similar triangles have to be congruent?

Stretch

  1. Only if they share a vertex
  2. No, their scale may differ from 1
  3. Yes always
  4. Only if acute
Hint

Similarity allows enlargement.

Answer and reasoning

No, their scale may differ from 1. Congruence additionally requires scale factor 1.

7. Does AAA determine equal size?

Foundation

  1. Yes always
  2. Only for obtuse triangles
  3. Only with labelled vertices
  4. No
Hint

Equal angles allow scaling.

Answer and reasoning

No. AAA determines similarity, while the scale factor can vary.

8. A similarity scale of 3 changes area by what factor?

Core

  1. 27
  2. 9
  3. 3
  4. 6
Hint

Square the length factor.

Answer and reasoning

9. Both base and altitude are multiplied by 3, giving factor 9.

9. Why is the included angle specified in SAS similarity?

Stretch

  1. The two proportional sides must enclose the matched angle
  2. Any unmatched angle works
  3. Angles do not matter
  4. It is only a naming convention
Hint

SSA can admit ambiguous triangles.

Answer and reasoning

The two proportional sides must enclose the matched angle. The included-angle condition makes the proportional-side construction determine the shape.

Choose your next step

Continue to Pythagoras, medians and Stewart’s theorem. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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