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# The Figure Shows a Circle with Centre O. Ab is the Side of Regular Pentagon and Ac is the Side of Regular Hexagon. Find the Angles of Triangle Abc. - ICSE Class 10 - Mathematics

ConceptArc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle

#### Question

The figure shows a circle with centre O. AB is the side of regular pentagon and AC is the side of regular hexagon.
Find the angles of triangle ABC.

#### Solution

Join OA, OB and OC
Since AB is the side of a regular pentagon,
∠AOB = (360°)/5 = 72°
Again AC is the side of a regular hexagon,
∠AOC = (360°)/6 = 60°
But ∠AOB + ∠AOC + ∠BOC = 360° [Angles at a point]
⇒ 72° + 60° + ∠BOC = 360°
⇒ 132° + ∠BOC = 360°
⇒ ∠BOC = 360° -132°
⇒ ∠BOC = 228°
Now, Arc BC subtends ∠BOC at the centre and
∠BAC at the remaining part of the circle.

⇒ ∠BAC = 1/2 ∠BOC

⇒∠BAC = 1/2xx 228° = 114°
Similarly, we can prove that
⇒∠ABC = 1 /2∠AOC
⇒∠ABC = 1 /2 xx 60° = 30°  And
⇒∠ACB = 1/2 AOB
⇒ ∠ACB = 1/2 xx  72° = 36°
Thus, angles of the triangle are, 114°, 30° and 36°

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#### Video TutorialsVIEW ALL [1]

Solution The Figure Shows a Circle with Centre O. Ab is the Side of Regular Pentagon and Ac is the Side of Regular Hexagon. Find the Angles of Triangle Abc. Concept: Arc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle.
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