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Byju's Answer

Standard X

Mathematics

Relations Using Incircle and Outcircle

In figure, AB...

Question

In figure, AB and CD are two parallel tangents to a circle with center O. ST is tangent segment between the two parallel tangents touching the circle at Q. Show that $∠ S O T = 90^{\circ}$ . [4 MARKS]

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Solution

Concept: 1 Mark
Application: 3 Marks

Join OQ, OP and OR

For $△ P O S$ and $△ Q O S$

OP = OQ (radius)

OS is the common base

As the lengths of tangents drawn from an external point to a circle are equal.

PS = QS

$∴ △ P O S ≅ △ Q O S$

$∠ P O S = ∠ Q O S$

Similarly $△ R O T ≅ △ Q O T$

$∠ R O T = ∠ Q O T$

POR is a straight line.

$∴ ∠ P O S + ∠ Q O S + ∠ R O T + ∠ Q O T = 180^{\circ}$

$2 ∠ Q O S + 2 ∠ Q O T = 180^{\circ}$

$∠ Q O S + ∠ Q O T = 90^{\circ}$

$∠ S O T = 90^{\circ}$

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In figure, AB and CD are two parallel tangents to a circle with center O. ST is tangent segment between the two parallel tangents touching the circle at Q. Show that ∠SOT=90∘. [4 MARKS]

In figure, AB and CD are two parallel tangents to a circle with center O. ST is tangent segment between the two parallel tangents touching the circle at Q. Show that $∠ S O T = 90^{\circ}$ . [4 MARKS]