Q3 of 12 Page 164

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.

Given:

Radius of larger circle = 5 cm.


Radius of smaller circle = 3 cm.


Solution:


The diagram for the question is as follows:



AB is a tangent for smaller circle, and we know that tangent at any point is perpendicular to the radius through point of contact.


Hence, OP ⊥ AB


Also, AB is a chord for large circle, and we know perpendicular from center to the chord bisects the chord


Hence,



Now, In ΔOAP


By Pythagoras theorem,


OA2 = AP2 + OP2


AP2 = OA2 – OP2


Here, OA = radius of bigger circle = 5 cm


and OB = radius of smaller circle = 3 cm


AP2 = 52 – 32


AP2 = 25 – 9


AP2 = 16


∴ AP = 4 cm


Hence,


AB = 2 AP


= 2(4) = 8 cm


More from this chapter

All 12 →
1

Write true or false of each of the following statements. Also give reason for your answer.

(i) A tangent of a circle is that line which intersects the circle at two points.


(ii) A tangent XY touches a circle at a point P and Q is any other point on the tangent. If O is the centre of the circle then, OP = OQ.


(iii) Two tangent LM and XY are drawn respectively at two points P and Q on a circle. If PQ is the diameter than LM || XY.


(iv) The centre ‘O’ of a circle lies on another circle whose centre is ‘A’. If the circle with centre O passed through the points A and B such that AOB is a straight line, then the tangents drawn from the point B will pass through the points of intersection of the two points.

 2

Fill in the blanks:

(i) Through a point on a circle…………………… tangent lines can be drawn.


(ii) A line intersecting a circle in two points is called a ……………… .


(iii) A circle can have …………………. Parallel tangents at the most.


(iv) The common point of a tangent to a circle and the circle is called…………….. .

 4

The length of a tangent from a point at a distance of 10 cm from the centre of the circle is 4 cm. Find the radius of the circle.

 5

A circle with centre O touches all the four sides of a quadrilateral ABCD. If the point of contact divides AB in the ratio 3 : 1 and AB = 8 cm, then find the radius of the circle if it is given OA = 10 cm.