Mathematics · 09 / 23
Similar triangles and parallel-line ratios
Middle school · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
Similar triangles share angles and have proportional corresponding sides. Parallel lines crossing two secants give equal corresponding length ratios. Match vertex order before cross multiplication. Scale factor k multiplies lengths by k and areas by k².
These triangles are similar. Doubling every length multiplies area by four while preserving angles.
- 01Angles or parallels
- 02Matching sides
- 03Common ratio
- 04Length versus area
WORKED EXAMPLE
From principle to reasoning
Corresponding sides 4 and 7 become 6 and 10.5 because k=6/4=1.5.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
What happens to lengths and area at scale factor 3?
One hint
Area uses two dimensions.
Reveal and understand the solution
Lengths triple and area multiplies by 9. A side-2 square becomes side 6, with area changing from 4 to 36.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Similar triangles have a length scale factor of 3. Find the area factor.
A hint to get started
Area multiplies two lengths.
Step-by-step solution
- Every length is multiplied by 3.
- Area is multiplied by 3².
- The area factor is 9, not 3.
Which step did you miss? Retry without help, then explain why the method works.
In triangle ABC, M lies on AB, N on AC and MN ∥ BC. AM = 3, AB = 9, BC = 12. Find MN.
A hint to get started
Match corresponding sides.
Step-by-step solution
- AM/AB = MN/BC by similar triangles.
- 3/9 = MN/12.
- MN = 4 in the same length unit.
Which step did you miss? Retry without help, then explain why the method works.
Does AM/AB = AN/AC = 1/2 prove MN ∥ BC if M and N are not on the sides?
A hint to get started
The converse also needs alignments and a compatible order.
Step-by-step solution
- Numerical equality alone does not describe the geometry.
- A,M,B and A,N,C must be aligned in a compatible order.
- With those conditions, the converse establishes parallelism.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
Match corresponding sides. The converse also needs alignment and ordering conditions.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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