Constructing a 30-60-90 triangle

After doing this	Your work should look like this
We start with a line segment PQ that will be the hypotenuse of the triangle..
First, we construct a 30° angle on one end of the hypotenuse. See Constructing a 30° angle.
1.  Starting with the hypotenuse line PQ, set the compasses on P, and set its width to any convenient width.
2.  Draw a broad arc across PQ. Label the point where it crosses PQ as point S.
3.  Without changing the compasses' width, move the compasses to the point S. Draw a broad arc that crosses the first one. Label the point where the two arcs cross as point T.
4.  Again without changing the compasses' width, move the compasses to the point T, and draw an arc across the previous arc, creating point R.
5.  Draw a line from P through R and continue it on beyond R to create the second side of the triangle.
Next, we construct a 60° angle on the other end of the hypotenuse. See Constructing a 60° angle.
6.  Place the compasses on point Q and set it to any convenient width.
7.  Draw a broad arc across PQ on the same side as point R. Label the point where it crosses PQ as point A.
8.  With the compasses on A, draw a second arc, crossing the first arc at point B
9.  Draw a line from Q, through B and on to cross the line PR. Label the intersection point C.
10.  Done. The triangle PQC is a 30-60-90 triangle.

Other constructions pages on this site

• List of printable constructions worksheets

Lines

• Introduction to constructions
• Copy a line segment
• Sum of n line segments
• Difference of two line segments
• Perpendicular bisector of a line segment
• Perpendicular at a point on a line
• Perpendicular from a line through a point
• Perpendicular from endpoint of a ray
• Divide a segment into n equal parts
• Parallel line through a point (angle copy)
• Parallel line through a point (rhombus)
• Parallel line through a point (translation)

Angles

• Bisecting an angle
• Copy an angle
• Construct a 30° angle
• Construct a 45° angle
• Construct a 60° angle
• Construct a 90° angle (right angle)
• Sum of n angles
• Difference of two angles
• Supplementary angle
• Complementary angle
• Constructing  75°  105°  120°  135°  150° angles and more

Triangles

• Copy a triangle
• Isosceles triangle, given base and side
• Isosceles triangle, given base and altitude
• Isosceles triangle, given leg and apex angle
• Equilateral triangle
• 30-60-90 triangle, given the hypotenuse
• Triangle, given 3 sides (sss)
• Triangle, given one side and adjacent angles (asa)
• Triangle, given two angles and non-included side (aas)
• Triangle, given two sides and included angle (sas)
• Triangle medians
• Triangle midsegment
• Triangle altitude
• Triangle altitude (outside case)

Right triangles

• Right Triangle, given one leg and hypotenuse (HL)
• Right Triangle, given both legs (LL)
• Right Triangle, given hypotenuse and one angle (HA)
• Right Triangle, given one leg and one angle (LA)

Triangle Centers

• Triangle incenter
• Triangle circumcenter
• Triangle orthocenter
• Triangle centroid

Circles, Arcs and Ellipses

• Finding the center of a circle
• Circle given 3 points
• Tangent at a point on the circle
• Tangents through an external point
• Tangents to two circles (external)
• Tangents to two circles (internal)
• Incircle of a triangle
• Focus points of a given ellipse
• Circumcircle of a triangle

Polygons

• Square given one side
• Square inscribed in a circle
• Hexagon given one side
• Hexagon inscribed in a given circle
• Pentagon inscribed in a given circle

Non-Euclidean constructions

• Construct an ellipse with string and pins
• Find the center of a circle with any right-angled object