This page shows how to construct a right triangle that has the hypotenuse (H) and one angle (A) given. It works in three steps:

- Copy the angle A. (See Copying an angle)
- Copy the length of the hypotenuse onto the angle leg (See Copying a segment)
- Drop a perpendicular from the end of the hypotenuse. (See Perpendicular to a line from a point)

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.

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

We first prove that ∆BCA is a right triangle | ||

1 | m∠BCA = 90° | BC was constructed using the procedure in Perpendicular to a line from a point. See that page for proof. |

2 | Therefore ∆BCA is a right triangle | By definition of a right triangle, one angle must be 90° |

Now prove BA is the hypotenuse H | ||

3 | AB = the given hypotenuse H | AB was copied from H at the same compass width |

Now prove ∠BAC is the given angle A | ||

4 | m∠BAC = given m∠A | Copied using the procedure in Copying an angle. See that page for proof |

9 | ∆BCA is a right triangle with the desired hypotenuse H and angle A | From (2), (3), (4) |

- 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 from a line at a point
- 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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