Question

In figure there are two concentric circles with centre O and radii 5cm and 3cm. from an external part P tangents PA and PB are drawn to these circles. If ${\text{AP}} = 12{\text{cm}}$ find the length of BP.

Answer 1

Hint: Draw a line from point P to point O, and then draw a perpendicular from point O to points A and B. The angle made by tangents to the centre is of right angle. So, we can apply Pythagoras theorem in it to solve our problem.

Complete step by step answer:
Given: we have given the radius of smaller circle is equal to $3{\text{cm}}$ and radius of large circle is equal to $5{\text{cm}}$.Length of tangent AP is equal to $12{\text{cm}}$ and we have to calculate the length of tangents BP.
First of all we have to join OA and OB and then join OP.

We know that the angles made of tangents to the centre of circle is equals to $90^\circ $ (right angle)
So, angle A and angle B are of $90^\circ $ .
Now, we take triangle AOP.
In triangle AOP
AP is equal to $12{\text{cm}}$(given)
OA is equal to $5{\text{cm}}$ (given)
Now, we calculate OP.
In the right angle triangle we applied Pythagoras theorem.
${{\text{H}}^{\text{2}}}{\text{ = }}{{\text{B}}^{\text{2}}}{\text{ + }}{{\text{P}}^{\text{2}}}$ where, ${\text{H}}$ is represented for hypotenuse, ${\text{B}}$ is represented for base, and ${\text{P}}$ is represented for perpendicular.
Now, substitute the values of base and perpendicular in the Pythagoras theorem.
$
  {\left( {{\text{OP}}} \right)^2} = {\left( {12} \right)^2} + {\left( 5 \right)^2} \\
  {\left( {{\text{OP}}} \right)^2} = 144 + 25 \\
  {\left( {{\text{OP}}} \right)^2} = 169 \\
  {\text{OP}} = \sqrt {169} \\
  {\text{OP}} = 13{\text{ cm}} \\
$
Now, we will take another triangle which is triangle BOP.
In triangle BOP,
BO is equal to $3{\text{ cm}}$ (given)
OP is equal to $13{\text{ cm}}$ (given)
So, by Pythagoras theorem we will find hypotenuse.
${\left( {OP} \right)^2} = {\left( {BO} \right)^2} + {\left( {BP} \right)^2}$
Substitute the values of BO and OP to find BP
$
  {\left( {13} \right)^2} = {\left( 3 \right)^2} + {\left( {{\text{BP}}} \right)^2} \\
  169 - 9 = {\left( {{\text{BP}}} \right)^2} \\
  160 = {\left( {{\text{BP}}} \right)^2} \\
  \sqrt {160} = {\text{BP}} \\
  12.65 = {\text{BP}} \\
$
So, the value of BP is equal to $12.65{\text{ cm}}$.
Note: We have given all the measurement angles of the triangle in the find of degree and take sides in the unit of cm. We applied the Pythagoras theorem on right angle triangles.