Exercises: Constructions 16 to 21

Describe the steps to construct an angle of $60^\circ$ using only a compass and straight edge.

Describe how to construct the centroid of a triangle $ABC$, naming the lines you draw.

Describe how to construct the tangent to a circle with centre $O$ at a given point $T$ on the circle.

Describe the construction of the circumcentre of a triangle $ABC$ and explain why the point found is the same distance from $A$, $B$ and $C$.

Describe the construction of the incentre of a triangle $ABC$, and state what is special about the circle centred at the incentre that touches all three sides.

Describe how to construct a parallelogram $ABCD$ given $|AB| = 6$ cm, $|AD| = 4$ cm and $|\angle DAB| = 60^\circ$, using compass and straight edge only.

A triangular park in Limerick has sides of 80 m, 100 m and 120 m. A fountain is to be placed at the point equidistant from the three corners.

1. a) Which point of the triangle is required and how is it constructed?
2. b) Explain why this point lies inside the triangle for this park, whereas for a triangle with an obtuse angle it lies outside.

Construct a $30^\circ$ angle without a protractor and justify each step. Then explain how to use this to construct a triangle $ABC$ with $|AB| = 7$ cm, $|\angle A| = 60^\circ$ and $|\angle B| = 30^\circ$, and state the size of $\angle C$ and the length $|AC|$ in terms of $|AB|$.

The centroid $G$ of a triangle $ABC$ divides each median in the ratio $2:1$, measured from the vertex.

1. a) Describe how to construct $G$.
2. b) For $A(0, 0)$, $B(6, 0)$ and $C(3, 9)$, find the coordinates of $G$.
3. c) Verify that $G$ lies two-thirds of the way along the median from $A$ to the midpoint of $[BC]$.