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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 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

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions