In the picture below, the radius of the circle is 15 cm. $∠ Q O R = 120 °$ . Select the statements that are true.

PR = 18√3 cm

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PQ = 18√3 cm

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PQ = 15√3 cm

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PR = 15√3 cm

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Solution

The correct options are
C
PQ = 15√3 cm

D
PR = 15√3 cm

Given, $∠ Q O R = 120 °$

Therefore, $∠ Q P R = 60 °$ [ $∠ Q O R + ∠ Q P R = 180 °$ ]

$∠ Q P O = 30 °$ [ $∠ Q P O = \frac{1}{2} ∠ Q P R$ ]

$∠ Q O P = 60 °$ [ $∠ Q O P + ∠ Q P O + ∠ O Q P = 180 °$ ]

In a right angled triangle where the angles are 30,60, 90, the sides are divided in the ratio 1:√3:2

Hence in △ POQ,

PO = 2 x OQ

= 2 x 15 = 30 cm

$P Q^{2} = 30^{2} - 15^{2}$

$P Q^{2} = 900 - 225$

$P Q^{2} = 675$

PQ = PR = 15√3 cm [Tangents from the same point are equal in length]

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In the picture below, the radius of the circle is 15 cm. ∠QOR=120°. Select the statements that are true.

In the picture below, the radius of the circle is 15 cm. $∠ Q O R = 120 °$ . Select the statements that are true.