This page shows how to construct a perpendicular at the end of a ray with compass and straightedge or ruler. This construction works as a result of Thales Theorem. From this theorem, we know that a diameter of a circle always subtends a right angle to any point on the circle, so by using it in reverse we produce a right angle.

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

- Create a circle that has the end part of the given ray as a chord. (step 3).
- We then draw a diameter of the circle (step 4).
- When we close the triangle in step 5, we know it must be a right triangle from Thales Theorem.

The image below is the final drawing from the animation above.

Argument | Reason | |
---|---|---|

1 | AB is the diameter of the circle center D. | A diameter is a line through the center of a circle. |

2 | m∠APB = 90° | From Thales Theorem - the diameter of a circle subtends a right angle to any point of the circle's circumference - here P. |

3 | AP is perpendicular to the endpoint of the ray PB | From (2) |

- Q.E.D

- 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)

- 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

- 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 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)

- 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

- 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

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