Name: Tangent and Chord for Concentric Circles
Uploaded: 2019-07-19
Duration: 2 min 18 s
Description: In an archery competition, the rings at which the archers aim are Concentric Circles. Can a line be Tangent to two concentric circles? Watch this video to le...

Question

Prove that if a chord is drawn from a point of contact of the tangent of the circle then angle made by this chord with the tangent are equal to the respective alternate angles made by segments with this chord.

Accepted Answer

If a line touches a circle and from the point of contact a chord is drawn, the angles which this chord makes with the given line are equal respectively to the angles formed in the corresponding alternate segments.

Given : The line $P Q$ (tangent) touching a circle with centre $O$ at point $A$ . From the point of contact $A$ , a chord $A B$ is drawn. $D$ and $C$ are the points in minor arc $A B$ and major arc $A B$ respectively of the circle.

Construction: Draw the diameter $A O B$ and join $E B$

Proof: (i) Since, at the point of contact, the angle between radius and tangent is $90 °$

$\Rightarrow ∠ E A Q = 90 °$

$\Rightarrow ∠ E A B + ∠ B A Q = 90 ° . . . . (1)$

In $△ A B E, ∠ A B E = 90 °$

$\Rightarrow ∠ E A B + ∠ A E B = 90 ° . . . (2)$

From $(1) & (2)$ , $∠ E A B + ∠ B A Q = ∠ E A B + ∠ A E B$

$\Rightarrow ∠ B A Q = ∠ A E B$

We know that, $∠ A E B = ∠ A C B$ (circle of segments are equal angles)

Therefore, $∠ B A Q = ∠ A C B$

Hence, proved.

Prove that if a chord is drawn from a point of contact of the tangent of the circle then angle made by this chord with the tangent are equal to the respective alternate angles made by segments with this chord.

Important Questions on Circles

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