Question

'O' is the center of the circle, AC is a tangent to a circle at point A. If $\Delta OAC$ is isosceles triangle then find the measure of $\angle OCA$
\[\begin{align}
  & A{{.75}^{\circ }} \\
 & B{{.50}^{\circ }} \\
 & C{{.45}^{\circ }} \\
 & D{{.40}^{\circ }} \\
\end{align}\]

Answer 1

Hint: In this question, we are given a circle with center 'O' having tangent at some point A. A triangle is formed named $\Delta OAC$ and we have to find one of the angles of the triangle. We will use the property that, radius of a circle drawn at that point to find one of the angles of the formed triangle. After that, we will use isosceles triangle property which states that angles opposite to corresponding equal sides are also equal. At last we will use angle sum property to find the required angle of the triangle.

Complete step-by-step answer:
Let us first draw diagram for better understanding:

Let O be the center of the circle and AC be the tangent to the circle. Join OA which is the radius of the circle. Also joint OC which forms triangle OAC.
As we know, the radius of the circle is perpendicular to the tangent at the circle. Hence $\angle OAC={{90}^{\circ }}$
Now, we have to find $\angle OCA$ let us suppose that $\angle OCA=x$
As we are given that $\Delta OCA$ is an isosceles triangle. Therefore, OA = AC
As we know, opposite angles of equal sides are also equal by isosceles triangle property. Therefore, $\angle AOC=\angle OCA$
As we have supposed $\angle OCA=x$ therefore, $\angle AOC=x$
Hence, in $\Delta OAC,\angle OAC={{90}^{\circ }},\angle OCA=x\text{ and }\angle AOC=x$
By using, sum of angles of triangle property which states that sum of the angles of a triangle is equal to ${{180}^{\circ }}$ we can say that:
\[\angle OAC+\angle OCA+\angle AOC={{180}^{\circ }}\]
Putting values of angles found earlier, we get:
\[\begin{align}
  & {{90}^{\circ }}+x+x={{180}^{\circ }} \\
 & {{90}^{\circ }}+2x={{180}^{\circ }} \\
 & 2x={{90}^{\circ }} \\
 & x={{45}^{\circ }} \\
\end{align}\]
But $\angle OCA=x$
Hence, \[\angle OCA={{45}^{\circ }}\] which is our required angle.

So, the correct answer is “Option C”.

Note: Students should always draw diagrams for these sums to avoid confusion and for better understanding. While taking equal sides in the isosceles triangle, we took OA = AC as we know OC can't be one of the equal sides because in the right angles triangle, it becomes hypotenuse and hypotenuse can never be equal to one of the sides of the triangle. Students should remember basic properties of circles as they can be used anywhere in any question.