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 diagonal bisection as a coordinate criterion.
- 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
For A=(0,0), B=(4,1), C=(6,4), D=(2,3), prove ABCD is a parallelogram.
Why this math matters
Use exact coordinate equalities to prove a shape property that a sketch only suggests. 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.
Prove a quadrilateral is a parallelogram with midpoints
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For A=(0,0), B=(4,1), C=(6,4), D=(2,3), prove ABCD is a parallelogram.
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
Midpoint of AC=((0+6)/2,(0+4)/2)=(3,2)
Average both coordinates of one diagonal's endpoints.
Develop the calculation
Midpoint of BD=((4+2)/2,(1+3)/2)=(3,2)
The other diagonal has the same midpoint.
Check and interpret
The diagonals bisect each other, so ABCD is a parallelogram
The given vertices form a nondegenerate quadrilateral; the converse diagonal-bisection theorem applies.
The result
The diagonals bisect each other, so ABCD is a parallelogram
The given vertices form a nondegenerate quadrilateral; the converse diagonal-bisection theorem applies.
Common mistakes to catch
- Compute the midpoint of each complete diagonal, not adjacent sides.
- A coordinate theorem needs a valid nondegenerate figure with the stated vertex order.
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 midpoint between (−5,7) and (3,−1).
Show a hint
Average each coordinate pair.
Reveal answer and explanation
(−1,3)
(−5+3)/2=−1 and (7−1)/2=3.
Practice 2
Do equal diagonal lengths alone prove a quadrilateral is a parallelogram?
Show a hint
Recall an isosceles trapezoid counterexample.
Reveal answer and explanation
No
Equal length and mutual bisection are different properties.
Take the idea with you
Use exact coordinate equalities to prove a shape property that a sketch only suggests.
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: Split a right triangle into similar triangles
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