Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 10 chapters and video availability01 · Read and understand
What you will learn
- Use the Pythagorean theorem for the distance between two planar points.
- 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
Find the distance between A=(−1,2) and B=(5,10).
Why this math matters
Choose a distance model appropriate to a straight route or a constrained grid route. 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 distance from horizontal and vertical changes
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the distance between A=(−1,2) and B=(5,10).
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
Δx=6; Δy=8
The coordinate changes form perpendicular legs of a right triangle.
Develop the calculation
d²=6²+8²=36+64=100
The connecting segment is the hypotenuse.
Check and interpret
d=10 units
Distance is the nonnegative square root and is unchanged if the endpoints are reversed.
The result
d=10 units
Distance is the nonnegative square root and is unchanged if the endpoints are reversed.
Common mistakes to catch
- Do not substitute a sum of absolute coordinate changes for Euclidean distance.
- Take the square root after summing squared changes.
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 (0,0) to (−3,−4).
Show a hint
Square both signed differences.
Reveal answer and explanation
5
√(9+16)=5; signs determine direction but squared lengths are positive.
Practice 2
Why is adding |Δx|+|Δy| not the straight-line distance in the main problem?
Show a hint
It follows a horizontal-then-vertical route.
Reveal answer and explanation
It gives a 14-unit grid route instead
The diagonal shortcut is ten units, while the two-leg path is fourteen.
Take the idea with you
Choose a distance model appropriate to a straight route or a constrained grid route.
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: Prove a quadrilateral is a parallelogram with midpoints
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