This page shows how to construct a right triangle that has the hypotenuse (H) and one leg (L) given. It is almost the same construction as Perpendicular at a point on a line, except the compass widths used are H and L instead of arbitrary widths.
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.
Multiple triangles possible
It is possible to draw more than one triangle has the side lengths as given. Youcan use the triangle to the left or right of the initial perpendicular, and also draw them below the initial line. All four are correct in that they satisfy the requirements, and are congruent to each other.
Proof
This construction works by effectively building two congruent triangles. The image below is the final drawing above with the blue line BP added
| Argument | Reason | |
|---|---|---|
| We first prove that ∆BCA is a right triangle | ||
| 1 | CP is congruent to CA | They were both drawn with the same compass width |
| 2 | BP is congruent to BA | They were both drawn with the same compass width |
| 3 | CB is common to both triangles BCP and BCA | Common side |
| 4 | Triangles ∆BCP and ∆BCA are congruent | Three sides congruent (SSS). |
| 5 | ∠BCP, ∠BCA are congruent | CPCTC. Corresponding parts of congruent triangles are congruent |
| 6 | m∠BCA = 90° | ∠BCA and ∠BCP are a linear pair and (so add to 180°) and congruent so each must be 90° |
| We now prove the triangle is the right size | ||
| 7 | CA is congruent to the given leg L | CA copied from L. See Copying a segment. |
| 8 | AB is congruent to the given hypotenuse H | Drawn with same compass width |
| 9 | ∆BCA is a right triangle with the desired side lengths | From (6), (7), (8) |
- Q.E.D
Try it yourself
Click here for a printable worksheet containing two AAS 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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