1. **Stating the problem:** We will explain four important theorems related to circles and tangents. 2. **Angle at centre = 2 × angle at circumference:** - This theorem states that the angle formed at the center of a circle by two points on the circle is twice the angle formed at the circumference by the same two points. - Formula: If $\angle AOB$ is the angle at the center and $\angle ACB$ is the angle at the circumference, then $$\angle AOB = 2 \times \angle ACB$$ - This is because the arc subtended by the angle at the center is twice the arc subtended at the circumference. 3. **Angle in semicircle = 90°:** - Any angle formed in a semicircle is a right angle. - If $AB$ is the diameter of the circle and $C$ is any point on the circle, then $$\angle ACB = 90^\circ$$ - This follows from the previous theorem since the angle at the center is $180^\circ$ (a straight line), so the angle at the circumference is half of that. 4. **Tangent ⟂ radius at point of contact:** - A tangent to a circle is perpendicular to the radius drawn to the point of contact. - If $T$ is the point of contact, then $$\angle OTB = 90^\circ$$ where $O$ is the center and $TB$ is the tangent. - This means the tangent line just touches the circle without cutting it. 5. **Tangents from same external point are equal:** - If two tangents are drawn from the same external point to a circle, their lengths are equal. - Let $P$ be the external point and $PA$, $PB$ be the tangents to the circle. - Then $$PA = PB$$ - This is because the two triangles formed by the radii and tangents are congruent by RHS (Right angle-Hypotenuse-Side) criterion. These theorems are fundamental in circle geometry and help solve many problems involving angles and tangents.