Constructing the tangent at a point on a circle

After doing this	Your work should look like this
We start with a point P somewhere on a given circle, with center point O. If the center is not given, you can use: "Finding the center of a circle with compass and straightedge or ruler", or "Finding the center of a circle with any right-angled object".
1.  Draw a straight line from the center O, through the given point P and on beyond P.
In the following steps 2 - 6 we are constructing the perpendicular to the line OP at a point P. This is the same procedure as described in Constructing a perpendicular at a point on a line.
2.  Put the compasses' point on P and set it to any width less than the distance OP. Then, on the line just drawn, draw an arc on each side of P. This creates the points Q and R as shown.
3.  Set the compasses on Q and set it to any width greater than the distance QP.
4.  Without changing the compasses' width, draw an arc approximately in the position shown on one side of P.
5.  Without changing the compasses' width, move the compasses to R and make another arc across the first, creating point S.
6.  Draw a line through P and S.
7.  Done. The line PS just drawn is the tangent to the circle O through point P.

Other constructions pages on this site

• List of printable constructions worksheets

Lines

• Introduction to constructions
• Copy a line segment
• Sum of n line segments
• Difference of two line segments
• Perpendicular bisector of a line segment
• Perpendicular at a point on a line
• Perpendicular from a line through a point
• Perpendicular from endpoint of a ray
• Divide a segment into n equal parts
• Parallel line through a point (angle copy)
• Parallel line through a point (rhombus)
• Parallel line through a point (translation)

Angles

• Bisecting an angle
• Copy an angle
• Construct a 30° angle
• Construct a 45° angle
• Construct a 60° angle
• Construct a 90° angle (right angle)
• Sum of n angles
• Difference of two angles
• Supplementary angle
• Complementary angle
• Constructing  75°  105°  120°  135°  150° angles and more

Triangles

• Copy a triangle
• Isosceles triangle, given base and side
• Isosceles triangle, given base and altitude
• Isosceles triangle, given leg and apex angle
• Equilateral triangle
• 30-60-90 triangle, given the hypotenuse
• Triangle, given 3 sides (sss)
• Triangle, given one side and adjacent angles (asa)
• Triangle, given two angles and non-included side (aas)
• Triangle, given two sides and included angle (sas)
• Triangle medians
• Triangle midsegment
• Triangle altitude
• Triangle altitude (outside case)

Right triangles

• Right Triangle, given one leg and hypotenuse (HL)
• Right Triangle, given both legs (LL)
• Right Triangle, given hypotenuse and one angle (HA)
• Right Triangle, given one leg and one angle (LA)

Triangle Centers

• Triangle incenter
• Triangle circumcenter
• Triangle orthocenter
• Triangle centroid

Circles, Arcs and Ellipses

• Finding the center of a circle
• Circle given 3 points
• Tangent at a point on the circle
• Tangents through an external point
• Tangents to two circles (external)
• Tangents to two circles (internal)
• Incircle of a triangle
• Focus points of a given ellipse
• Circumcircle of a triangle

Polygons

• Square given one side
• Square inscribed in a circle
• Hexagon given one side
• Hexagon inscribed in a given circle
• Pentagon inscribed in a given circle

Non-Euclidean constructions

• Construct an ellipse with string and pins
• Find the center of a circle with any right-angled object