Circles and Geometric Shapes | Exercise 5.6

Question 1

In a circle with centre OO, the central angle ∠AOB\angle AOB is 60∘60^\circ. If the radius of the circle is 12 cm12\text{ cm}, what is the length of the chord ABAB?

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Solution
Understand the Question
  • In ΔAOB\Delta AOB, OAOA and OBOB are radii of the circle, so OA=OB=12 cmOA = OB = 12\text{ cm}.
  • Since OA=OBOA = OB, ΔAOB\Delta AOB is an isosceles triangle with ∠OAB=∠OBA\angle OAB = \angle OBA.
  • Given the central angle ∠AOB=60∘\angle AOB = 60^\circ, we can find the remaining two angles using the angle sum property of a triangle.
  • Showing all angles equal 60∘60^\circ proves ΔAOB\Delta AOB is equilateral, meaning chord ABAB is equal in length to the radius.

Step 1 · Find the Base Angles of Triangle AOB

In ΔAOB\Delta AOB, OA=OB=12 cmOA = OB = 12\text{ cm} (radii of the same circle).Diagram 1

Since angles opposite to equal sides are equal: ∠OAB=∠OBA=x\angle OAB = \angle OBA = x

By the angle sum property in ΔAOB\Delta AOB:

∠AOB+∠OAB+∠OBA=180∘60∘+x+x=180∘60∘+2x=180∘2x=180∘−60∘2x=120∘x=120∘2=60∘\begin{aligned} \angle AOB + \angle OAB + \angle OBA &= 180^\circ \\ 60^\circ + x + x &= 180^\circ \\ 60^\circ + 2x &= 180^\circ \\ 2x &= 180^\circ - 60^\circ \\ 2x &= 120^\circ \\[0.6em] x &= \dfrac{120^\circ}{2} = 60^\circ \end{aligned}

Therefore, ∠OAB=60∘\angle OAB = 60^\circ and ∠OBA=60∘\angle OBA = 60^\circ.

Step 2 · Determine the Length of Chord AB

Since ∠AOB=∠OAB=∠OBA=60∘\angle AOB = \angle OAB = \angle OBA = 60^\circ, ΔAOB\Delta AOB is an equilateral triangle.

All sides of an equilateral triangle are equal: AB=OA=OB=12 cmAB = OA = OB = 12\text{ cm}

Answer

12 cm12\text{ cm}

Common Mistakes
  • Assuming Right Triangle: Mistakenly assuming ΔAOB\Delta AOB is a right-angled triangle and incorrectly applying the Pythagoras theorem.
  • Equilateral Property Confusion: Forgetting that an isosceles triangle with one 60∘60^\circ angle is always an equilateral triangle, where chord length directly equals the radius.

More questions in Exercise 5.6

Q1

In a circle with centre OO, the central angle ∠AOB\angle AOB is 60∘60^\circ. If the radius of the circle is 12 cm12\text{ cm}, what is the length of the chord ABAB?

Q2

Let AA and BB be two points on a circle with centre OO.

(i) Are there points X,YX, Y on the circle, on the same side of ABAB, such that ∠AXB\angle AXB is different from ∠AYB\angle AYB?

(ii) Is it true that if ∠AXB=∠AYB\angle AXB = \angle AYB, then XX and YY lie on the same side of the circle?

(iii) If ∠AXB=∠AYB\angle AXB = \angle AYB, and XX and YY do not lie on the circle, does the circle through AA, BB and XX also pass through YY?

Q3

Find xx in Fig. 5.26.

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