

Right triangle given each leg (LL)
Geometry construction using a compass and straightedge
This page shows how to construct a
right triangle that has both the given leg lengths.
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.
You can 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 lines PQ and QA 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 
PQ is congruent to AQ 
They were both drawn with the same compass width 
3 
CQ is common to both triangles ∆PQC and ∆AQC 
Common side 
4 
Triangles ∆PQC and ∆AQC are
congruent 
Three sides congruent (SSS). 
5 
∠QCP, ∠QCA are congruent 
CPCTC. Corresponding parts of congruent triangles are congruent 
6 
m∠QCA = 90° 
∠QCA and ∠QCP are a
linear pair and (so add to 180°)
and congruent so each must be 90° 
7 
∆BCA is a right triangle 
∠BCA = 90°. 
We now prove the triangle is the right size 
8 
CA is congruent to the given leg L1 
CA copied from L1. See
Copying a segment. 
9 
BC is congruent to the given leg L2 
Drawn with same compass width 
10 
∆BCA is a right triangle with the desired side lengths 
(7), (8), (9) 
 Q.E.D
Try it yourself
Click here for a printable worksheet containing two LL 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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