# Find the length of the tangent drawn from a point whose distance from the center of a circle is 25 cm. Given that the radius of the circle is 7 cm.

Last updated date: 24th Mar 2023

•

Total views: 306.6k

•

Views today: 7.83k

Answer

Verified

306.6k+ views

Hint: Draw a rough figure of the tangent meeting the circle. The tangent to a circle is always perpendicular to the radius through the point of contact. You will get a right angled triangle. Solve it and find the length of the tangent drawn.

Complete step-by-step answer:

Let us consider ‘O’ as the center of the circle. Consider the figure drawn.

Let OT be the radius of the circle.

\[\therefore \]OT = 7 cm.

Let P be the point from which the tangent is drawn to the circle. The tangent meets at point T on the circle. Given that the length from point P to the center O of the circle is 25 cm.

\[\therefore \]Length of OP = 25 cm.

What we need to find is the length of PT.

From the figure, we can assume that radius OT is perpendicular to the tangent drawn, i.e. the tangent to a circle is always perpendicular to the radius through the point of contact.

\[\therefore \angle OTP={{90}^{\circ }}\]

Now let us consider the right angled triangle OTP.

By basic geometry we know that,

\[O{{P}^{2}}=P{{T}^{2}}+O{{T}^{2}}\]i.e.\[{{\left( hypotenuse \right)}^{2}}={{\left( altitude \right)}^{2}}+{{\left( base \right)}^{2}}\]

\[\begin{align}

& \therefore P{{T}^{2}}=O{{P}^{2}}-O{{T}^{2}} \\

& PT=\sqrt{O{{P}^{2}}-O{{T}^{2}}} \\

& PT=\sqrt{{{25}^{2}}-{{7}^{2}}}=\sqrt{625-49}=24cm \\

\end{align}\]

Hence, the length of tangent from point P = 24 cm.

Note: There are a lot of special properties for a tangent to a circle, like a tangent can never cross the circle. It can only touch the circle, like how we have drawn in the figure. At the point of tangency, it is perpendicular to the radius. Therefore, we took angle OTP as \[{{90}^{\circ }}\]. So when you get a question related to tangents, remember both the points.

Complete step-by-step answer:

Let us consider ‘O’ as the center of the circle. Consider the figure drawn.

Let OT be the radius of the circle.

\[\therefore \]OT = 7 cm.

Let P be the point from which the tangent is drawn to the circle. The tangent meets at point T on the circle. Given that the length from point P to the center O of the circle is 25 cm.

\[\therefore \]Length of OP = 25 cm.

What we need to find is the length of PT.

From the figure, we can assume that radius OT is perpendicular to the tangent drawn, i.e. the tangent to a circle is always perpendicular to the radius through the point of contact.

\[\therefore \angle OTP={{90}^{\circ }}\]

Now let us consider the right angled triangle OTP.

By basic geometry we know that,

\[O{{P}^{2}}=P{{T}^{2}}+O{{T}^{2}}\]i.e.\[{{\left( hypotenuse \right)}^{2}}={{\left( altitude \right)}^{2}}+{{\left( base \right)}^{2}}\]

\[\begin{align}

& \therefore P{{T}^{2}}=O{{P}^{2}}-O{{T}^{2}} \\

& PT=\sqrt{O{{P}^{2}}-O{{T}^{2}}} \\

& PT=\sqrt{{{25}^{2}}-{{7}^{2}}}=\sqrt{625-49}=24cm \\

\end{align}\]

Hence, the length of tangent from point P = 24 cm.

Note: There are a lot of special properties for a tangent to a circle, like a tangent can never cross the circle. It can only touch the circle, like how we have drawn in the figure. At the point of tangency, it is perpendicular to the radius. Therefore, we took angle OTP as \[{{90}^{\circ }}\]. So when you get a question related to tangents, remember both the points.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE