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Teaching video
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Grade 10 chapters and video availability01 · Read and understand
What you will learn
- Explain why two reflections in intersecting lines produce a rotation by twice their oriented angle.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Algebraic equations, ratios, angle and area facts, and basic coordinate geometry.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Reflect an arbitrary point (x,y) across the x-axis and then the y-axis. Prove the resulting motion is a half-turn and compare this with mirrors thirty degrees apart.
Why this math matters
Use the angle rule to predict a composition of reflections without testing every vertex separately. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use Euclidean geometry and the angle, parallelism, similarity, or congruence conditions stated; a sketch alone does not establish them.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Derive a rotation from two intersecting reflections
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Reflect an arbitrary point (x,y) across the x-axis and then the y-axis. Prove the resulting motion is a half-turn and compare this with mirrors thirty degrees apart.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
First reflection: (x,y)→(x,−y)
Reflection reverses the coordinate perpendicular to its mirror line.
Develop the calculation
Second reflection: (x,−y)→(−x,−y)
Every displacement from the origin is reversed, proving a half-turn for all points rather than just one example.
Check and interpret
The composition is a 180° rotation about the origin
More generally, reflecting a direction angle α across a line at angle β gives 2β−α. Two mirrors at angles β and γ therefore send α to α+2(γ−β), a rotation by twice the oriented mirror angle.
The result
The composition is a 180° rotation about the origin
More generally, reflecting a direction angle α across a line at angle β gives 2β−α. Two mirrors at angles β and γ therefore send α to α+2(γ−β), a rotation by twice the oriented mirror angle.
Common mistakes to catch
- The rotation center is the intersection of the two mirror lines.
- The order-independence of perpendicular reflections does not hold for every pair of intersecting lines.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Does reversing the order of the two coordinate-axis reflections change the final image?
Show a hint
Compare a positive and negative half-turn.
Reveal answer and explanation
No; both produce (−x,−y)
Rotations by 180° and −180° have the same final effect; perpendicular mirrors are a special case.
Practice 2
Reflect (1,0) first across the x-axis, then across the line through the origin at 30°. What if the order is reversed?
Show a hint
The two orders give rotations of 60° and −60°.
Reveal answer and explanation
First order: (1/2,√3/2); reverse order: (1/2,−√3/2)
The mirrors are not perpendicular, so opposite signed rotations give different images; composition order matters.
Take the idea with you
Use the angle rule to predict a composition of reflections without testing every vertex separately.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Two parallel reflections create a translation
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