Tangent To Circle At Point

 

ON Circle

In a circle, a radius drawn to the point of tangency is perpendicular to the tangent.

Construct a perpendicular to a line from a point on the line. 

Given: Circle O

Construct: a tangent to circle at a point on the circle

STEPS:

1. If a point is given on the circle, connect the point to the center of the circle to form a radius. If a point on the circle is not given, then draw any radius and label P.

2. Extend the radius out of  the circle.

3. Construct a perpendicular line to the radius line at point P.

OUTSIDE CIRCLE

Imagine in your mind what these tangents to a circle will look like. The diagram below shows two tangents from point P. we need not draw the two tangents, but both will remind you of how this construction will look, as the construction creates two possible tangents.

Given: Circle O

Construct: a tangent to circle O from P

STEPS:

1. Connect O to P.

2. Construct bisector of.

3. Place compass point at midpoint of

and stretch span to O or P.

4. Draw circle.

5. Connect P to where the two circles intersect to create tangents.

When the construction is finished, connect O to A to form a radius of circle O.

This radius also forms ΔOAP in circle M. Sinceis the diameter of circle M, ∠OAP is an angle inscribed in a semicircle, making it a right angle.

Since ∠OAP is a right angle, . In circle O, we now have the radius perpendicular to a line passing through a point on the circle (A), makinga tangent to circle O.

FINDING CENTRE OF THE CIRCLE

What do you do when a construction problem involving circles, gives you a starting circle such as that shown at the right?

There is NO CENTER indicated on the circle. Unfortunately, you can NOT plot your best guess of where you think the center may be located.

If you encounter this situation, you will have to CONSTRUCT the location of the center.

Given: Circle with no center indicated

Construct: locate the center of the circle

STEPS:

1. Draw an inscribed angle (an angle with its vertex on the circle and sides terminating on the circle).

(This construction also works if you draw two chords instead of the inscribed angle. Drawing the ∠ keeps the chords positioned to more clearly find the center.)

2. Bisect each side of the angle (or chord).

3. The point where the bisectors intersect is the center of the circle.

As per the theorem, "in a circle, the perpendicular bisector of a chord passes through the center of the circle".

The diagram shown above supports this theorem with congruent triangles.