Question

How do you find $x$ when line $AB$, $CD$ and $AD$ are tangent of circle $O$ ? (Point $O$ is the center of the circle, line $BO$ and $OC$ are the radius of circle $O$ .)

Answer 1

Hint: In the given figure three tangents are drawn on a circle. Two of them are drawn at the edges of a diameter and the third one intersects them both. We can use the two-tangent theorem to find the value of $AD$. Further we have to use the properties of similarity of triangles to find the value of $x$ using known lengths of segments.

Complete step by step solution:
We have in the figure three tangents $AB$, $CD$ and $AD$ drawn on a circle. We have to find the value of $x$, i.e. the length of the segment $MN$.

We know the length of $AB = 3\;units$ and $CD = 7\;units$.
From two-tangent theorem, we know that if two tangents are drawn to one circle from the same external point, then their lengths are equal.
In the given figure, we can see that $AB$ and $AM$ are drawn from the same point $A$.
Thus, $AB = AM = 3\;units$
Similarly, we can see that $CD$ and $DM$ are drawn from the same point $D$.
Thus, $CD = DM = 7\;units$
$AD = AM + DM = 3 + 7 = 10\;units$
Since $AB\parallel CD$, $\angle BAN = \angle DCN,\;\angle ABN = \angle CDN,\;\angle CND = \angle ANB$.
Thus, $\Delta ANB$ is similar to $\Delta CND$.
The ratio of corresponding sides of similar triangles are equal.
Thus, $\dfrac{{AN}}{{CN}} = \dfrac{{AB}}{{CD}} = \dfrac{3}{7}$
Now in $\Delta CAD$ and $\Delta NAM$, \[\dfrac{{AN}}{{CN}} = \dfrac{3}{7}\] and $\dfrac{{AM}}{{DM}} = \dfrac{3}{7}$. Therefore, $\Delta CAD$ is similar to $\Delta NAM$ and $MN\parallel CD$.
\[\dfrac{{MN}}{{CD}} = \dfrac{{AN}}{{AC}} = \dfrac{{AM}}{{AD}} = \dfrac{3}{{10}}\]
Therefore, \[MN = \dfrac{3}{{10}} \times CD = \dfrac{3}{{10}} \times 7 = \dfrac{{21}}{{10}} = 2.1\;units\]
Hence, the value of $x$ is \[2.1\;units\].

Note: We used a two-tangent theorem and similarity of triangles to solve the problem. For using a two-tangent theorem, both the tangents must be drawn from a single point. When using properties of similarity of triangles, care should be taken to mark corresponding segments and angles of the triangles. Although we could have calculated the radius, we didn’t require it to solve the problem.