Name: ___KEY_________________
Paper Folding Constructions
This activity uses folding of square pieces of paper, compass, ruler, and protractor to
construct the bisector of an angle, perpendicular bisector of a line segment, incenter of a
triangle, circumcenter of a triangle, and centroid of a triangle.
Work in pairs. One person should read the directions and the other should do the folding.
Change roles for each construction. The questions you must answer are in bold.
I.
Construct the angle bisector.
4. Measure ∠ ABD and ∠ CBD. What is true about the measures of ∠ ABD and
∠ CBD? What is the relationship between BD and ∠ ABC?
!!!"
BD is the angle bisector of ∠ ABC.
5. Compare DF and DE. What did you find? What is the relationship between
the circle and the sides of ∠ABC?
DF and DE are equal. The sides are tangent to the circle.
II.
Construct the perpendicular bisector of a line segment.
2. Fold P onto Q and crease the paper. Open the paper and mark the point M. M is the
_midpoint_ of PQ.
3. Mark a point R on the crease. Measure ∠ RMP and ∠ RMQ. What is the measure
of each angle?
90°
4. Describe the relationship between RM and PQ .
They are perpendicular.
5. Draw RP and RQ . Fold the paper on RM again. What is true about RP and RQ?
They are equal.
6. Choose any other point X on RM . What is true about PX and QX?
They are equal.
7. What can be said about any point X on RM and its relationship to P and Q?
It is equidistant from P and Q.
III.
Construct the incenter of a triangle.
2. What appears to be true about the three angle bisectors?
They intersect in a single point.
IV.
Construction the circumcenter of a triangle.
2. Fold the paper to construct the perpendicular bisector of each side of the triangle.
What appears to be true about the perpendicular bisectors?
They intersect in a single point.
3. Label the point of intersection Q. Measure DQ, EQ, and FQ. What did you find?
They are equal.
5. Repeat the steps above for an obtuse triangle and a right triangle. How are the
results the same? How are they different?
They intersect in a single point or they are inscribed in the circle. Either is
acceptable.
They are not always inside the triangle.
V.
Construct the centroid of a triangle.
2. Draw the medians XC , YB , and ZA . What appears to be true about the medians?
They intersect in a single point.
3. Label the point of intersection, called the centroid, M. Measure the two segments
on each median and find the ratio of the length of the shorter segment to the length
of the longer segment. The centroid divides each median into two segments so that
the ratio of the lengths is ___1 : 2___________.