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)
Printable step-by-step instructions
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.
Proof
| 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
Try it yourself
Click here for a printable worksheet containing two HA triangle construction problems. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.Other constructions pages on this site
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
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
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