Circles | Exercise 10.2

Question 4

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

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Solution

Diameter: A chord passing through the centre of a circle. Its two endpoints lie on opposite ends of the circle, and it is the longest chord.

A radius is always perpendicular to the tangent at the point of contact.

Step 1 — Draw the setup

Let's draw a circle. Let its center be O. Let AB be a diameter of this circle. Let RS be the tangent line at point A. Let PQ be the tangent line at point B.

Diagram 1

Step 2 — Apply tangent-radius property

Tangent-Radius Perpendicularity Theorem: The radius drawn to the point of tangency is always perpendicular to the tangent at that point.

We know the radius is perpendicular to the tangent. Radius OA touches tangent RS at point A. So, OA is perpendicular to RS. This means the angle ∠OAR\angle OAR is 90 degrees. ∠OAR=90∘\angle OAR = 90^\circ Similarly, radius OB touches tangent PQ at point B. So, OB is perpendicular to PQ. This means the angle ∠OBQ\angle OBQ is 90 degrees. ∠OBQ=90∘\angle OBQ = 90^\circ

Both angles are 90∘\boxed{\text{Both angles are } 90^\circ}

Step 3 — Identify parallel lines

Let's consider lines RS and PQ. Let AB be the transversal line. From Step 2, we found ∠OAR=90∘\angle OAR = 90^\circ. We also found ∠OBQ=90∘\angle OBQ = 90^\circ. So, these two angles are equal. ∠OAR=∠OBQ\angle OAR = \angle OBQ Alternate Interior Angles: When a transversal crosses two lines, the angles formed on opposite sides of the transversal between the two lines are called alternate interior angles. If these angles are equal, the two lines are parallel.

These angles are alternate interior angles. When alternate interior angles are equal, the lines are parallel. Therefore, line RS is parallel to line PQ.

Answer

(i) The tangents drawn at the ends of a diameter of a circle are parallel.

More questions in Exercise 10.2

Q1

In Q.1 to 3, choose the correct option and give justification.

  1. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is (A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm
Q2

In Q.1 to 3, choose the correct option and give justification.

  1. In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ=110∘\angle \text{POQ} = 110^\circ, then ∠PTQ\angle \text{PTQ} is equal to (A) 60∘60^\circ (B) 70∘70^\circ (C) 80∘80^\circ (D) 90∘90^\circ
Q3

In Q.1 to 3, choose the correct option and give justification.

  1. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80∘80^\circ, then ∠POA\angle \text{POA} is equal to (A) 50∘50^\circ (B) 60∘60^\circ (C) 70∘70^\circ (D) 80∘80^\circ
Q4

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Q5

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Q6

The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Q7

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Q8

A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that

AB+CD=AD+BC\text{AB} + \text{CD} = \text{AD} + \text{BC}

Q9

In Fig. 10.13, XY and X′Y′\text{X}'\text{Y}' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′\text{X}'\text{Y}' at B. Prove that ∠AOB=90∘\angle\text{AOB} = 90^\circ.

Q10

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Q11

Prove that the parallelogram circumscribing a circle is a rhombus.

Q12

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.

Q13

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

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