Constructions 16 to 21
A council in Limerick wants to put a fountain in a triangular park so that it is exactly the same distance from all three corners. There is no GPS and no measuring tape, only string, pegs and a straight line. Which point of the triangle is it, and how do you find it? By the end of this page you will be able to answer that, using only a compass and a straight edge.
Rules of the game and a 60° angle
In a construction you use only a compass and a straight edge (an unmarked ruler). You may not measure angles with a protractor. You keep your construction arcs on the page, because they are the evidence for your steps.
Two ideas come up again and again. The first is that a compass arc is the set of points a fixed distance from a centre. The second is that two arcs with the same radius meet at points that are the same distance from both centres.
Construction 18: an angle of 60° without a protractor. The idea is to build an equilateral triangle. All its angles are 60°.
Describe the steps to construct an angle of $60^\circ$ using only a compass and straight edge.
Describe the steps to construct an angle of $60^\circ$ using only a compass and straight edge.
Circumcentre and incentre
Every triangle has special points. Two of them come from bisectors.
Construction 16: the circumcentre. The circumcentre is where the perpendicular bisectors of the sides meet.
Construction 17: the incentre. The incentre is where the angle bisectors meet.
Back to the park: the fountain must be at the circumcentre, because that is the point equidistant from the three corners. Construct the perpendicular bisectors of two sides.
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 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 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.
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. a) Which point of the…
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.
- a) Which point of the triangle is required and how is it constructed?
- b) Explain why this point lies inside the triangle for this park, whereas for a triangle with an obtuse angle it lies outside.
Tangent to a circle and a parallelogram
Construction 19: the tangent at a point $T$ on a circle. A tangent touches the circle at one point and is perpendicular to the radius at that point (Theorem 20).
Construction 20: a parallelogram. Opposite sides of a parallelogram are equal and parallel. So you need the angle, two sides and then a fourth vertex.
Describe how to construct the tangent to a circle with centre $O$ at a given point $T$ on the circle.
Describe how to construct the tangent to a circle with centre $O$ at a given point $T$ on the circle.
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.
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.
Centroid and a 30° angle
Construction 21: the centroid. A median joins a vertex to the midpoint of the opposite side. The three medians meet at the centroid $G$.
A 30° angle. Construct 60° and then bisect it. Bisecting an angle of 60° gives $60^\circ \div 2 = 30^\circ$. Each step is justified: the 60° from the equilateral triangle, the bisector from equal arcs.
Describe how to construct the centroid of a triangle $ABC$, naming the lines you draw.
Describe how to construct the centroid of a triangle $ABC$, naming the lines you draw.
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,…
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. a) Describe how to construct $G$. b) For $A(0, 0)$, $B(6, 0)$ and…
The centroid $G$ of a triangle $ABC$ divides each median in the ratio $2:1$, measured from the vertex.
- a) Describe how to construct $G$.
- b) For $A(0, 0)$, $B(6, 0)$ and $C(3, 9)$, find the coordinates of $G$.
- c) Verify that $G$ lies two-thirds of the way along the median from $A$ to the midpoint of $[BC]$.