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Geometry Level 3

The given figure shows sector OAB with centre O and radius 54cm.Another circle XYZ with centre P,is enclosed by sector OAB.If angle AOB =60.Find the area of OXPY.


The answer is 561.2.

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1 solution

Curtis Clement
Oct 22, 2015

O, P and Z are collinear so O Z = 54 ⟹ O P = O Z − P Z = 54 − r \ OZ = 54 \implies\ OP = OZ - PZ = 54 - r Now by congruent triangles OPY and OPX: ∠ P O Y = 1 2 X O Y = 30 a n d O P Y = 60 \angle POY = \frac{1}{2} XOY = 30 \ ~ \ and \ OPY = 60 ∴ c o s 60 = 1 2 = r 54 − r ⟹ r = 18 \therefore\ cos60 = \frac{1}{2} = \frac{r}{54-r} \implies\ r = 18 ∴ [ O X P Y ] = a b s i n C = 18 × 36 s i n 60 = 324 3 \therefore\ [OXPY] = absinC = 18 \times\ 36 sin60 = 324 \sqrt{3}

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A minor correction: The last line should have 36 instead of 38

Krutarth Patel - 5 years, 6 months ago

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ahh well spotted :)

Curtis Clement - 5 years, 6 months ago

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