# In the adjoining figure, $CBA$ is secant and $CD$ is tangent to the circle. If $AB=7cm$ and

$BC=9cm$, then find the length of $CD$.

Last updated date: 16th Mar 2023

•

Total views: 306.3k

•

Views today: 4.85k

Answer

Verified

306.3k+ views

Hint: We can equate the ratio of corresponding sides of similar triangles if we can prove that any two triangles that contain the sides $AC$, $BC$ and $CD$ are similar triangles. By equating this ratio, we can get a relation between these three sides.

In the above figure, let us consider $\Delta ADC$and $\Delta DBC$.

From the figure, we can see that the angle $C$ is common in both triangles. So, we can write

$\angle C=\angle C........\left( i \right)$

Also, there is a property of the circle which is related to the tangent and chord. This property states that

“the angle between the chord and the tangent is equal to the angle made by the chord in the alternative

segment”. Using this property in the above figure in $\Delta ADC$and $\Delta DBC$, we get

$\angle DAC=\angle BDC.......\left( ii \right)$

From equation $\left( i \right)$ and equation $\left( ii \right)$ , using $AA$ similarity rule, we get that

the triangles are similar.

$\Delta ADC\sim \Delta DBC$

Since these two triangles are similar, we can equate the ratio of corresponding sides of the

corresponding triangles. This means that we can equate the following sides,

$\dfrac{BC}{DC}=\dfrac{DC}{AC}$

$\Rightarrow {{\left( DC \right)}^{2}}=AC\times BC........(iii)$

It is given in the question that side $BC=9cm$.

We can see from the figure that $ABC$ is a straight line. So, we can obtain a relation as below,

$AC=AB+BC........(iv)$

Substituting values of $AB=7cm$ and $BC=9cm$ in equation $(iv)$, we get

$AC=7+9$

$\Rightarrow AB=7cm$

Substituting the values of $AC=16cm$ and $BC=9cm$ in equation $\left( iii \right)$, we get

${{\left( DC \right)}^{2}}=16\times 9$

$\Rightarrow DC=\sqrt{16\times 9}$

$\Rightarrow DC=\sqrt{16}\times \sqrt{9}$

$\Rightarrow DC=4\times 3$

$\Rightarrow DC=12cm$

$\Rightarrow DC=CD=12cm$

Note: In this question, we have to find a relation between the sides $AC$ and $BC$, whose lengths are already given in the question, so as to obtain the required side $CD$. That is why we have to prove similarity of the triangles to obtain a relation between the length of these sides.

In the above figure, let us consider $\Delta ADC$and $\Delta DBC$.

From the figure, we can see that the angle $C$ is common in both triangles. So, we can write

$\angle C=\angle C........\left( i \right)$

Also, there is a property of the circle which is related to the tangent and chord. This property states that

“the angle between the chord and the tangent is equal to the angle made by the chord in the alternative

segment”. Using this property in the above figure in $\Delta ADC$and $\Delta DBC$, we get

$\angle DAC=\angle BDC.......\left( ii \right)$

From equation $\left( i \right)$ and equation $\left( ii \right)$ , using $AA$ similarity rule, we get that

the triangles are similar.

$\Delta ADC\sim \Delta DBC$

Since these two triangles are similar, we can equate the ratio of corresponding sides of the

corresponding triangles. This means that we can equate the following sides,

$\dfrac{BC}{DC}=\dfrac{DC}{AC}$

$\Rightarrow {{\left( DC \right)}^{2}}=AC\times BC........(iii)$

It is given in the question that side $BC=9cm$.

We can see from the figure that $ABC$ is a straight line. So, we can obtain a relation as below,

$AC=AB+BC........(iv)$

Substituting values of $AB=7cm$ and $BC=9cm$ in equation $(iv)$, we get

$AC=7+9$

$\Rightarrow AB=7cm$

Substituting the values of $AC=16cm$ and $BC=9cm$ in equation $\left( iii \right)$, we get

${{\left( DC \right)}^{2}}=16\times 9$

$\Rightarrow DC=\sqrt{16\times 9}$

$\Rightarrow DC=\sqrt{16}\times \sqrt{9}$

$\Rightarrow DC=4\times 3$

$\Rightarrow DC=12cm$

$\Rightarrow DC=CD=12cm$

Note: In this question, we have to find a relation between the sides $AC$ and $BC$, whose lengths are already given in the question, so as to obtain the required side $CD$. That is why we have to prove similarity of the triangles to obtain a relation between the length of these sides.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

The coordinates of the points A and B are a0 and a0 class 11 maths JEE_Main

Trending doubts

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

Tropic of Cancer passes through how many states? Name them.

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

What is per capita income

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

A Short Paragraph on our Country India