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 8 chapters and video availability01 · Read and understand
What you will learn
- Explain how to test whether three sides form a right triangle.
- Solve the two practice problems and explain how the assumptions affect the answers.
Before you start
Squares and the triangle inequality.
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
A triangle has side lengths 9 cm, 12 cm, and 15 cm. Determine whether it is a right triangle.
Why this math matters
The converse of the Pythagorean theorem recognizes a right triangle from a square-sum equality. Use exact or appropriately toleranced measurements before treating a near equality as a geometric proof.
Set up the model
A useful answer starts with clear assumptions:
- The lengths are exact positive measurements in a Euclidean plane.
- The three lengths describe one triangle.
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.
Test whether three sides form a right triangle
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A triangle has side lengths 9 cm, 12 cm, and 15 cm. Determine whether it is a right triangle.
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 quantities
largest side c = 15; 9 + 12 > 15
The longest side is the only possible hypotenuse; the lengths also satisfy the triangle inequality.
Apply the relationship
9² + 12² = 81 + 144 = 225; 15² = 225
Compare the sum of the two smaller squares with the largest square.
Check and interpret
The triangle is right, with hypotenuse 15 cm
Equality establishes a right angle opposite the longest side.
The result
The triangle is right, with hypotenuse 15 cm
Equality establishes a right angle opposite the longest side.
Common mistakes to catch
- The largest length must be the candidate hypotenuse.
- A square-sum test does not replace checking that a valid triangle exists.
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
Are side lengths 5, 6, and 8 a right triangle?
Show a hint
Compare 5² + 6² with 8².
Reveal answer and explanation
No
25 + 36 = 61, which differs from 64.
Practice 2
Do lengths 1, 2, and 3 form a triangle to which a right-angle test can apply?
Show a hint
Check the strict triangle inequality first.
Reveal answer and explanation
No nondegenerate triangle
1 + 2 = 3 creates a straight, collapsed configuration rather than a triangle with positive area.
Take the idea with you
Use exact or appropriately toleranced measurements before treating a near equality as a geometric proof.
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: Turn coordinate differences into a distance
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