Department of Mathematics Education

EMT 669, Jim Wilson

### Triangle Construction Problems

In these problems, information is given about a triangle to identify physical
attributes. The problem is to describe ruler and compass constructions of
the triangle from the given attributes. For consistent notation adapted
to this environment I use the following:
The angles at vertices A, B, C: **A, B, C**

The sides of the triangle opposite vertices A, B, C: **a, b, c.**

The medians: **m(a), m(b), m(c)**

The altitudes: **h(a), h(b), h(c)**

The segments along an angle bisector to the opposite side: **d(a), d(b),
d(c)**

The radius of the circumcircle: **R**

The radius of the incircle: **r**

The radii of the excircles: **r(a), r(b), r(c)**

The notation m(a), etc., is usually written with a subscript and I will
do so in the sketches. The notation: **b+c **will mean "given a
line segment equal in lenght to the sum of the lengths of sides **b**
and **c**.

1. Triangle, given two sides and the median to the third side.
# b, c, m(a)

**One Solution** (In Microsoft Word)

**A GSP Sketch**

A GSP Script

2. Triangle,
given two angles and the perimeter
# A, B, a+b+c

3 . Triangle, given the three medians
# m(a), m(b), m(c)

4. Triangle, given two sides and the angle opposite one of the sides
# a,b,A

5. Triangle, given two sides and the altitude to the third
# a,c, h(b)

6. Triangle, given a side, the median to that side, and another median
# b,m(b), m(a)

7. Triangle, given a side and the angle subtending (opposite of) that
side, and a segment equal to the sum of the other two sides.
# a, b+c, A

8. Triangle, given a side and the angle bisecting segments to the sides
opposite each endpoint of the given segment.
# a, d(b), d(c)

9. Triangle, given the perimeter, an angle, and the altitude from that
angle.
# a+b+c, A, h(a)

10. Triangle, given a side, an angle adjacent to the side, and a segment
equal to the sum of the other two sides.
# a, B, b+c

11.
# b+c, B, h(c)

12.
# b+c, C, h(b)

13

# b+c, A, B

14.
# b+c, a, h(b)