Answer

Verified

435.3k+ views

Hint: Let us draw a figure using the given conditions in the question. So, that it will be clearer what should be the tangent to the given circle.

Complete step-by-step answer:

So, as we can see from the figure drawn above that OP is the radius of the circle.

And OP = 9 cm.

And Q is the point from which the distance to the centre of the circle is 15 cm (OQ = 15 cm).

So, now PQ will be the tangent drawn from point Q to the circle.

And as we know that from the property of the tangent drawn from any external point to the circles forms a right-angle triangle, with hypotenuse as the line joining the centre of the circle with the external point and the other two sides are radius of circle and the length of tangent .

So, here OPQ will be a right-angled triangle, right-angled at P (point where tangent touches the circle).

Now we had to find the value of PQ.

So, we can apply Pythagoras theorem in the triangle OPQ.

According to Pythagoras theorem, if ABC is any right-angled triangle, right-angled at B. And AC is the hypotenuse of triangle ABC.

Then, \[{\left( {{\text{AC}}} \right)^{\text{2}}}{\text{ = }}{\left( {{\text{AB}}} \right)^{\text{2}}}{\text{ + }}{\left( {{\text{BC}}} \right)^{\text{2}}}\]

So, applying Pythagoras theorem in triangle OPQ. We get,

\[{\left( {{\text{OQ}}} \right)^{\text{2}}}{\text{ = }}{\left( {{\text{OP}}} \right)^{\text{2}}}{\text{ + }}{\left( {{\text{PQ}}} \right)^{\text{2}}}\]

So, \[{\left( {{\text{PQ}}} \right)^{\text{2}}}{\text{ = }}{\left( {{\text{OQ}}} \right)^{\text{2}}}{\text{ - }}{\left( {{\text{OP}}} \right)^{\text{2}}}\]

Now as we can see from the above figure that OP = 9 cm and OQ = 15 cm.

So, \[{\left( {{\text{PQ}}} \right)^{\text{2}}}{\text{ = }}{\left( {15} \right)^{\text{2}}}{\text{ - }}{\left( 9 \right)^{\text{2}}}{\text{ = 225 - 81 = 144}}\]

So, \[{\left( {{\text{PQ}}} \right)^{\text{2}}}{\text{ = 144}}\]

Now take the square root on both sides of the above equation. We get,

PQ = 12 cm (because length cannot be negative).

Hence, the length of the tangent drawn from a point which is 15 cm away from the centre of the circle is 12 cm.

Note: Whenever we come up with this type of problem where we are asked to find the length of the tangent from an external point then we should use the Pythagoras theorem because tangents drawn from the external point to the circle form a right-angled triangle.

Complete step-by-step answer:

So, as we can see from the figure drawn above that OP is the radius of the circle.

And OP = 9 cm.

And Q is the point from which the distance to the centre of the circle is 15 cm (OQ = 15 cm).

So, now PQ will be the tangent drawn from point Q to the circle.

And as we know that from the property of the tangent drawn from any external point to the circles forms a right-angle triangle, with hypotenuse as the line joining the centre of the circle with the external point and the other two sides are radius of circle and the length of tangent .

So, here OPQ will be a right-angled triangle, right-angled at P (point where tangent touches the circle).

Now we had to find the value of PQ.

So, we can apply Pythagoras theorem in the triangle OPQ.

According to Pythagoras theorem, if ABC is any right-angled triangle, right-angled at B. And AC is the hypotenuse of triangle ABC.

Then, \[{\left( {{\text{AC}}} \right)^{\text{2}}}{\text{ = }}{\left( {{\text{AB}}} \right)^{\text{2}}}{\text{ + }}{\left( {{\text{BC}}} \right)^{\text{2}}}\]

So, applying Pythagoras theorem in triangle OPQ. We get,

\[{\left( {{\text{OQ}}} \right)^{\text{2}}}{\text{ = }}{\left( {{\text{OP}}} \right)^{\text{2}}}{\text{ + }}{\left( {{\text{PQ}}} \right)^{\text{2}}}\]

So, \[{\left( {{\text{PQ}}} \right)^{\text{2}}}{\text{ = }}{\left( {{\text{OQ}}} \right)^{\text{2}}}{\text{ - }}{\left( {{\text{OP}}} \right)^{\text{2}}}\]

Now as we can see from the above figure that OP = 9 cm and OQ = 15 cm.

So, \[{\left( {{\text{PQ}}} \right)^{\text{2}}}{\text{ = }}{\left( {15} \right)^{\text{2}}}{\text{ - }}{\left( 9 \right)^{\text{2}}}{\text{ = 225 - 81 = 144}}\]

So, \[{\left( {{\text{PQ}}} \right)^{\text{2}}}{\text{ = 144}}\]

Now take the square root on both sides of the above equation. We get,

PQ = 12 cm (because length cannot be negative).

Hence, the length of the tangent drawn from a point which is 15 cm away from the centre of the circle is 12 cm.

Note: Whenever we come up with this type of problem where we are asked to find the length of the tangent from an external point then we should use the Pythagoras theorem because tangents drawn from the external point to the circle form a right-angled triangle.

Recently Updated Pages

Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred

The branch of science which deals with nature and natural class 10 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Trending doubts

What was the Metternich system and how did it provide class 11 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What organs are located on the left side of your body class 11 biology CBSE

What is pollution? How many types of pollution? Define it