NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 9: Circles - Exercise 9.4

Question 4

Step 1: Look at the figure. AB is a chord of the circle, AOC is a diameter, and AT is a tangent at point A.

Step 2: Recall the tangent–chord theorem (also called the Alternate Segment Theorem). It says: The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment of the circle.

Step 3: Here, the tangent is AT, and the chord through the point of contact is AB.

Step 4: So, the angle between AT and AB is \(\angle BAT\).

Step 5: According to the theorem, this angle must be equal to the angle subtended by the same chord AB in the alternate segment of the circle.

Step 6: The chord AB subtends an angle at point C on the opposite segment of the circle, which is \(\angle ACB\).

Step 7: Therefore, by the tangent–chord theorem, we get: \[ \angle BAT = \angle ACB \]

Final Result: Hence proved that \(\angle BAT = \angle ACB\).