Circles and Geometric Shapes | Exercise 5.4

Question 1

Use the Baudhāyana–Pythagoras theorem to show why Theorem 6 must be true.

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Solution
Understand the Question
  • Theorem 6: Chords of equal length in a circle are equidistant from the center.
  • The perpendicular drawn from the center of a circle to a chord bisects the chord.
  • Joining the center to the endpoints of the chords forms right-angled triangles with the radii as hypotenuses.
  • Applying the Baudhāyana–Pythagoras theorem to both triangles allows us to prove that the perpendicular distances from the center to the chords are equal.

Step 1 · Set up the Geometry and Relate Chord Halves

Consider a circle with center OO and two equal chords AB=CDAB = CD.

Draw perpendiculars OM⊥ABOM \perp AB and ON⊥CDON \perp CD.Diagram 1

Since the perpendicular from the center bisects a chord: AM=AB2AM = \dfrac{AB}{2} CN=CD2CN = \dfrac{CD}{2}

Given AB=CDAB = CD: AM=CNAM = CN

Also, OAOA and OCOC are radii of the same circle: OA=OCOA = OC

Step 2 · Apply Baudhāyana–Pythagoras Theorem

In right-angled ΔOMA\Delta OMA: OA2=OM2+AM2OA^2 = OM^2 + AM^2

In right-angled ΔONC\Delta ONC: OC2=ON2+CN2OC^2 = ON^2 + CN^2

Since OA=OCOA = OC: OA2=OC2OA^2 = OC^2

OM2+AM2=ON2+CN2OM^2 + AM^2 = ON^2 + CN^2

Substitute CN=AMCN = AM:

OM2+AM2=ON2+AM2OM2=ON2OM=ON\begin{aligned} OM^2 + AM^2 &= ON^2 + AM^2 \\ OM^2 &= ON^2 \\ OM &= ON \end{aligned}
Answer

Since OM=ONOM = ON, equal chords are equidistant from the center of the circle.

Common Mistakes
  • Perpendicular Distance: The distance of a chord from the center is strictly the perpendicular segment from the center to the chord.
  • Bisector Property: Forgetting to state that the perpendicular from the center bisects the chord, which is necessary to equate AM=CNAM = CN from AB=CDAB = CD.

More questions in Exercise 5.4

Q1

Use the Baudhāyana–Pythagoras theorem to show why Theorem 6 must be true.

Q2

Consider Fig. 5.15. If CECE is perpendicular to ABAB, CHCH is perpendicular to GHGH, and CE=CHCE = CH, show that AB=GFAB = GF.

Q3

Solve the previous question using the Baudhāyana–Pythagoras theorem.

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