Lessons 35 · 1 lessons · A. A. Rahimqoriyev, M. A. To‘xtaxo‘jayeva. Geometry, Grade 8 (textbook for general secondary schools), revised 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2014
35
Properties of a chord and a diameter
Textbook: pp. 109–110
GoalKnow the property of a diameter perpendicular to a chord and the fact that no chord exceeds the diameter.
New words
midpoint of a chord · vatar o‘rtasidistance from the centre to a chord · markazdan vatargacha masofaperpendicular bisector · o‘rta perpendikulyarisosceles triangle · teng yonli uchburchak
Explanation
Theorem: a diameter perpendicular to a chord bisects the chord and the arc it subtends. Proof: since OA = OB = R, triangle AOB is isosceles, and the perpendicular OP from the centre to the chord is its altitude, median and bisector. Hence AP = PB and ∠AOP = ∠BOP, so the arcs are equal too. Second theorem: no chord is longer than the diameter, because in the right triangle OPB the leg PB is shorter than the hypotenuse OB, so 2PB ≤ 2OB. Corollaries: a diameter through the midpoint of a chord is perpendicular to it; the perpendicular bisector of a chord passes through the centre. If the radius is R and the distance from the centre to the chord is d, then (half the chord)² = R² − d².
Worked examples
R = 13, chord 24: half the chord is 12, and d² = 169 − 144 = 25, so d = 5.
R = 10, d = 6: half the chord squared is 100 − 36 = 64, half the chord is 8, so the chord is 16.
Class activity
“Midpoint of a chord”: cut a paper circle (with an adult’s help), fold it along a chord and check that the fold line through the chord’s midpoint passes through the centre.
Practice
1
In a circle of radius 17 a chord is 30 long. What is the distance from the centre to the chord?
8
2
The radius is 5 and the distance from the centre to the chord is 3. How long is the chord?
8
3
A diameter is perpendicular to a chord of length 18. How long are the parts it divides the chord into (each)?
9
4
Why does a diameter perpendicular to a chord bisect it?
Triangle AOB is isosceles and OP is its altitude, so it is also its median.