MathematicsChapter 9: Chord and Arcs of a Circle
Pythagoras in the Circle (skill)
Radius, half-chord, and distance from centre make a right triangle.
Learning objectives
- 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.
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 a semicircle is 90 deg (often listed with angles of a circle; the right triangle is the same skill).
Mathematical form
m = 5
d = 12 cm
r = 13 cm
chord = 10 cm. Find d. m
Where / symbols
- 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.
Overview
Label r, d, m:
The Three Lengths
r2 = d2 + m2 with m half the chord. Given any two, find the third. Radius 13, half 5, d=12 is the 5-12-13 set.
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.
Examples Angle in semicircle 90 · Hyp is the diameter · Not a random chord as hyp
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 r2 = d2 + m2 169 = d2 + 25 d2 = 144 d = 12 cm Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- r = 13 cm, chord = 10 cm. Find d.
- m = 5
- r2 = d2 + m2
- 169 = d2 + 25
- d2 = 144
- d = 12 cm
In simple words
Use half the chord, not the full chord, in Pythagoras with d and r.
Key points
- r, d, half-chord
- right triangle
- r2 = d2 + m2
- diameter is hyp
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Pythagoras in the Circle (skill)
Sticky Notes · Quick Revision
Core Idea
- 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
- r2 = d2 + m2 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.
Worked Method
- Worked: r = 13 cm, chord = 10 cm. Find d.
- m = 5
- r2 = d2 + m2
- 169 = d2 + 25
- d2 = 144
- d = 12 cm
Common Misconception
- Use half the chord, not the full chord, in Pythagoras with d and r.
FBISE Exam Check
- Use half the chord, not the full chord, in Pythagoras with d and r.
60-second revision
- 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
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