Geometric Constructions

Using the center of the circle, Without using the center of the circle, Drawing a perpendicular to the radius at the point

Step 1: Method 1 uses the center to draw a radius and then constructs a perpendicular at the point. Step 2: Method 2 uses the tangent-secant angle theorem without locating the center. Step 3: Both methods rely on the fundamental property that a tangent is perpendicular to the radius at the point of contact.

True

Step 1: From any external point, exactly two tangent lines can be drawn to a circle. Step 2: This is a fundamental theorem in geometry. Step 3: The two tangents from an external point are equal in length and make equal angles with the line joining the external point to the center.

Compass, Ruler (straight edge)

Step 1: A compass is needed to draw arcs for construction. Step 2: A ruler (straight edge) is needed to draw straight lines. Step 3: Tangent construction uses basic geometric construction techniques that only require compass and straight edge. Step 4: Protractor and set square are not essential for this construction.

Angle inscribed in semicircle is 90°

Step 1: The construction uses the property that any angle inscribed in a semicircle is a right angle. Step 2: When the auxiliary circle intersects the original circle, it creates right angles at those intersection points. Step 3: Since tangent is perpendicular to radius, these intersection points are the required tangent points. Step 4: Lines from P through these points are the required tangents.