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Pythagoras in the Circle (skill)
Radius, half-chord, and distance from centre make a right triangle. Almost every numerical chord question is this one triangle. Diameter as hypotenuse: angle in a semicircle is 90 deg (often listed with angles of a circle; the right triangle is the same skill).
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Pythagoras in the Circle (skill)
What is it?
Radius, half-chord, and distance from centre make a right triangle. Almost every numerical chord question is this one triangle. Diameter as hypotenuse: angle in a semicircle is 90 deg (often listed with angles of a circle; the right triangle is the same skill).
In simple words
- Radius, half-chord, and distance from centre make a right triangle.
- THE THREE LENGTHS: r^2 = d^2 + m^2 with m half the chord.
- DIAMETER HINT: If a triangle is inscribed with one side a diameter, the angle at the rim is 90 deg, so Pythagoras applies on that triangle too.
- Worked-method reminder: d = 12 cm
Important terms
PYTHAGORAS IN THE CIRCLE (SKILL): Radius, half-chord, and distance from centre make a right triangle. Almost every numerical chord question is this one triangle. Diameter as hypotenuse: angle in a semicircle is 90 deg (often listed with angles of a circle; the right triangle is the same skill).
THE THREE LENGTHS: r^2 = d^2 + m^2 with m half the chord. Given any two, find the third. Radius 13, half 5, d=12 is the 5-12-13 set.
DIAMETER HINT: If a triangle is inscribed with one side a diameter, the angle at the rim is 90 deg, so Pythagoras applies on that triangle too. That theorem is named with the angle chapter; the calculation is still a^2+b^2=c^2.
What you should learn
- Explain pythagoras in the circle (skill) using the correct definitions, conditions, and notation.
- Apply r, d, half-chord, right triangle correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- m = half chord
- d = distance from O
- r = OA
THE THREE LENGTHS
r^2 = d^2 + m^2 with m half the chord. Given any two, find the third. Radius 13, half 5, d=12 is the 5-12-13 set.
DIAMETER HINT
If a triangle is inscribed with one side a diameter, the angle at the rim is 90 deg, so Pythagoras applies on that triangle too. That theorem is named with the angle chapter; the calculation is still a^2+b^2=c^2.
Recommended for Class 10
Full school-level explanation
Radius, half-chord, and distance from centre make a right triangle. Almost every numerical chord question is this one triangle. Diameter as hypotenuse: angle in a semicircle is 90 deg (often listed with angles of a circle; the right triangle is the same skill).
Label r, d, m:
WHAT — PYTHAGORAS IN THE CIRCLE (SKILL)
Radius, half-chord, and distance from centre make a right triangle. Almost every numerical chord question is this one triangle. Diameter as hypotenuse: angle in a semicircle is 90 deg (often listed with angles of a circle; the right triangle is the same skill). The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Pythagoras In The Circle (Skill) matters because chord and arc theorems convert circle diagrams into equal lengths, equal angles, and right triangles. They are the foundation for later tangent proofs and constructions.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through THE THREE LENGTHS, then DIAMETER HINT. Worked: r = 13 cm, chord = 10 cm. Find d. m = 5 r^2 = d^2 + m^2 169 = d^2 + 25 d^2 = 144 d = 12 cm Finish by checking signs, restrictions, units, and whether the answer matches the question.
THE THREE LENGTHS
r^2 = d^2 + m^2 with m half the chord. Given any two, find the third. Radius 13, half 5, d=12 is the 5-12-13 set.
Remember
- r2 = d2 + m2
Examples
- m = half chord
- d = distance from O
- r = OA
DIAMETER HINT
If a triangle is inscribed with one side a diameter, the angle at the rim is 90 deg, so Pythagoras applies on that triangle too. That theorem is named with the angle chapter; the calculation is still a^2+b^2=c^2.
Remember
- diameter is hyp
Examples
- Angle in semicircle 90
- Hyp is the diameter
- Not a random chord as hyp
Remember
- r, d, half-chord
- right triangle
- d = 12 cm
- half-chord m
- r2 = d2 + m2
- diameter is hyp
- r, d, half-chord
- right triangle
How the ideas connect
Worked: r = 13 cm, chord = 10 cm. Find d. m = 5 r2 = d2 + m2 169 = d2 + 25 d2 = 144 d = 12 cm
Deeper understanding
Go beyond memorising Pythagoras in the Circle (skill). Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: r = 13 cm, chord = 10 cm. Find d. m = 5 r2 = d2 + m2 169 = d2 + 25 d2 = 144 d = 12 cm
Common mistakes
Exam focus
Core Idea
THE THREE LENGTHS
DIAMETER HINT
Worked Method
Exam focus
Quick revision
- r, d, half-chord
- right triangle
- d = 12 cm
- half-chord m
- PYTHAGORAS IN THE CIRCLE (SKILL): Radius, half-chord, and distance from centre make a right triangle. Almost every numerical chord que
- THE THREE LENGTHS: r^2 = d^2 + m^2 with m half the chord. Given any two, find the third. Radius 13, half 5, d=12 is the
Use half the chord, not the full chord, in Pythagoras with d and r.
For long answers on Pythagoras in the Circle (skill): define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- r, d, half-chord
- right triangle
- d = 12 cm
- half-chord m
Important formulas
- r, d, half-chord
- right triangle
- d = 12 cm
Common exam traps
- Use half the chord, not the full chord, in Pythagoras with d and r.
- Not a random chord as hyp
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Use half the chord, not the full chord, in Pythagoras with d and r.
- Mistake
Not a random chord as hyp
- Mistake
Do not treat THE THREE LENGTHS and DIAMETER HINT as the same step; each has a different purpose.
- Mistake
Use half the chord, not the full chord, in Pythagoras with d and r.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
r, d, half-chord
right triangle
d = 12 cm
half-chord m
Radius, half-chord, and distance from centre make a right triangle. Almost every numerical chord question is this one triangle. Diameter as hypotenuse: angle in
d = 12 cm
- Definition
- Radius, half-chord, and distance from centre make a right triangle. Almost every numerical chord question is this one triangle. Diameter as hypotenuse: angle in
- Formula
- d = 12 cm
- Example
- m = half chord
- Critical warning
- Use half the chord, not the full chord, in Pythagoras with d and r.
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Exam Practice
Define
Q1. Define Pythagoras in the Circle (skill).
Explain
Q2. Explain: THE THREE LENGTHS.
Example
Q3. Give one clear example of Pythagoras in the Circle (skill) and say why it fits.
Mistake
Q4. State one common mistake students make about Pythagoras in the Circle (skill).
Exam note
Q5. Write a short exam note on Pythagoras in the Circle (skill): definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.