Learn with Amar
Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Find displacement components.
- Apply the distance formula.
- Check a midpoint geometrically.
Before you start
Coordinate pairs, squares, square roots, and averages.
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
Two points on a square coordinate map are A = (2, 1) and B = (8, 9), in metres. Find their straight-line separation and midpoint.
Why this math matters
Coordinates encode both horizontal and vertical changes. A diagonal route uses both changes at once, while a route following grid streets adds separate horizontal and vertical legs.
Set up the model
A useful answer starts with clear assumptions:
- The axes are perpendicular and use the same metre scale.
- Straight travel between points is allowed.
- The map is a flat Euclidean model.
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.
Find a straight route and its halfway point
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Two points on a square coordinate map are A = (2, 1) and B = (8, 9), in metres. Find their straight-line separation and midpoint.
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.
Subtract corresponding coordinates
Δx = 8 − 2 = 6 m; Δy = 9 − 1 = 8 m
These perpendicular changes form the legs of a right triangle whose hypotenuse connects A and B.
Combine the changes
d = √(6² + 8²) = √100 = 10 m
Adding 6 and 8 would describe a two-leg grid route of 14 metres, not this direct diagonal.
Average and check
M = ((2 + 8)/2, (1 + 9)/2) = (5, 5)
From either endpoint to M, the coordinate changes have magnitudes 3 and 4. Each half therefore has length 5 metres.
The result
The direct distance is 10 m and the midpoint is (5, 5).
The midpoint lies on the connecting segment. Matching horizontal and vertical averages ensures it divides both displacement components equally.
Common mistakes to catch
- The distance formula needs perpendicular axes with a consistent scale.
- Averaging x and y within one point does not produce a midpoint.
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
Find the distance from (−1, 2) to (2, 6).
Show a hint
The coordinate changes are 3 and 4.
Reveal answer and explanation
5 units
√(3² + 4²) = 5.
Practice 2
A segment begins at (2, −1) and has midpoint (5, 3). Find its other endpoint.
Show a hint
Double the midpoint, then subtract the known endpoint.
Reveal answer and explanation
(8, 7)
The missing coordinates are 2(5) − 2 = 8 and 2(3) − (−1) = 7.
Take the idea with you
Separate straight-line distance from restricted routes when reading a grid-based diagram.
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: Move a design with reflections, rotations, and translations
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