In the adjoining figure, two circles with centres A and B, and of radii 5 cm and 3 cm touch each other internally. If the perpendicular bisector of AB meets the bigger circle in P and Q, find the length of PQ.

Question

In the adjoining figure, two circles with centres A and B, and of radii 5 cm and 3 cm touch each other internally. If the perpendicular bisector of AB meets the bigger circle in P and Q, find the length of PQ.

Filo · Accepted Answer

It is given that two circles with centers at

A

and

B

and of radii

5 c m

and

3 c m

touch each other internally.

The perpendicular bisector of

A B

meets the bigger circle in

P

Join the line

A P

From the figure, we know that

PQ

intersects the line

A B

at

L

So we get

A B = (5 - 3)

A B = 2 c m

We know that

PQ

is the perpendicular bisector of

A B

So we get

A L = \frac{1}{2} \times A B

Substituting the values

A L = \frac{1}{2} \times 2

So we get

A L = 1 c m

Consider

△ P L A

Using the Pythagoras theorem

A P^{2} = A L^{2} + P L^{2}

By substituting the values we get

5^{2} = 1^{2} + P L^{2}

So we get

P L^{2} = 5^{2} - 1^{2}

On further calculation

P L^{2} = 25 - 1

P L^{2} = 24

By taking the square root

P L = 24

So we get

P L = 26 c m

We know that

PQ = 2 \times P L

By substituting the value

PQ = 2 \times 26

PQ = 46 c m

Therefore, then length of

PQ = 46 c m

Question Text	In the adjoining figure, two circles with centres $A$ and $B$ , and of radii $5 c m$ and $3 c m$ touch each other internally. If the perpendicular bisector of $A B$ meets the bigger circle in $P$ and $Q$ , find the length of $PQ$ .
Answer Type	Text solution:1
Upvotes	150

In the adjoining figure, two circles with centres $A$ and $B$ , and of radii $5 c m$ and $3 c m$ touch each other internally. If the perpendicular bisector of $A B$ meets the bigger circle in $P$ and $Q$ , find the length of $PQ$ .

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