# Right triangle given each leg (LL)

This page shows how to construct a right triangle that has both the given leg lengths.

## 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. 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.