Isosceles triangle given base and altitude
Geometry construction using a compass and straightedge

How to draw an isosceles triangle given the base and altitude with compass and straightedge or ruler. The base is the unequal side of the triangle and the altitude is the perpendicular height from the base to the apex. It works by first copying the base segment, then constructing its perpendicular bisector. The apex is then marked up from the base.

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.

Proof

The image below is the final drawing from the above animation.

Argument Reason
1 PR = CD By construction. PR is a copy of the segment CD. See Copying a line segment for method and proof.
2 QS is the perpendicular bisector of PR By construction. See Constructing the perpendicular bisector of a line segment..
3 Triangles QPS and QRS are congruent. SAS. See Test for congruence side-angle-side.
• QS common to both
• ∠QSR = ∠QSP = 90° from (2).
• SP = SR from (2).
4 QP = QR CPCTC - Corresponding Parts of Congruent Triangles are Congruent
5 QS = AB By construction.
6 QPR is an isosceles triangle. From (4). An isosceles triangle has two sides the same length.
7 QPR is an isosceles triangle with base CD and altitude AB. From (1) (5) (6)

- Q.E.D

Try it yourself

Click here for a printable isosceles construction worksheet containing two problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at   patreon.com/mathopenref