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Is the following statement true or false?

The line segment that joins the points of contact of two parallel tangents of a circle passes through its center.

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True

False

answer is A.

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Detailed Solution

Given a statement, the line segment that joins the points of contact of two parallel tangents of a circle passes through its center.
Angle sum property of triangle states that the sum of interior angles of a triangle is

180 °

.
Let

X B Y

and

P C Q

be two parallel tangents to a circle with center

O

.
We join

OB

and

OC

. Then draw

O A | | X Y

Sum of adjacent interior angles is

180 °

XB | | AO ⇒ ∠ XBO + ∠ AOB = 180 °

∠ XBO = 90 °

(Radius is perpendicular to the tangent at the point of contact.)

⇒ 90 ° + ∠ A O B = 180 ° ⇒ ∠ A O B = 180 ° − 90 ° = 90 °

After repeating the same process we get,

∠ A O C = 90 ° ∴ ∠ AOB + ∠ AOC = 90 ° + 90 ° = 180 °

Therefore

BOC

is a straight line passing through

O

which is true.
Therefore the correct option is 1.

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