Question

Line BC is tangent to the circle centered at A. Find the measure of angle BCA. Explain or show your reasoning.

Answer 1

The measure of angle BCA is 90 degrees because a tangent to a circle is perpendicular to the radius at the point of tangency, forming a right triangle.

Step-by-step explanation:

The measure of angle BCA in the context provided where line BC is tangent to a circle centered at A is 90 degrees. When a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency. Consequently, since line BC is tangent to the circle at point B, and assuming that the radius from the center A to the point B is also line AB, triangle ABC would be a right triangle with angle BCA as the right angle.

Answer 2

`m\angle BCA= 33\degree`

Step-by-step explanation:

`In\: \odot A` BC is tangent to the circle at point B.

Therefore, by tangent-radius theorem:

`\therefore AB\perp BC`

`\therefore \angle ABC = 90\degree`

`In\: \triangle ABC,`

`m\angle ABC +m\angle BAC +m\angle BCA= 180\degree`

`90\degree+57\degree +m\angle BCA= 180\degree`

`147\degree +m\angle BCA= 180\degree`

`m\angle BCA= 180\degree - 147\degree`

`m\angle BCA= 33\degree`