

Right triangle given one leg and hypotenuse (HL)
Geometry construction using a compass and straightedge
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 stepbystep instructions
The above animation is available as a
printable stepbystep 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.
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Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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