Applied Mathematics Gaps for Junior Cycle: Trigonometry, Probability, Arithmetic, Sets and Algebra · Transformations and Constructions
Constructions 4 to 15
A groundskeeper at a GAA club needs to mark out a small rectangular training area. She has pegs and a long rope, but no set square and no tape. How can she get four perfect square corners using only the rope as a "compass"? By the end of this page you will know the constructions that do exactly this, and why they work.
Perpendiculars
In a construction you only use the instruments the question allows. "Using only a compass and straight edge" means the ruler is for drawing straight lines only, so you must not use it to measure. When a question says "construct a triangle with sides 6 cm, 5 cm and 4 cm", you may use the ruler to set the compass to those lengths.
You already know two basic constructions:
Construction 1: the bisector of an angle.
Construction 2: the perpendicular bisector of a line segment $[AB]$.
Now two more. A line $\ell$ and a point $P$ give two cases: $P$ is on the line, or $P$ is not on it.
Practice★☆☆Not started
Using only a compass and straight edge, draw a line segment $[AB]$ of length $7 \text{ cm}$. Construct the perpendicular bisector of $[AB]$ and state which construction…
Using only a compass and straight edge, draw a line segment $[AB]$ of length $7 \text{ cm}$. Construct the perpendicular bisector of $[AB]$ and state which construction number this is.
Two constructions let you put a given measurement exactly where you want it.
You can also make a $60^\circ$ angle with a compass alone.
Why $60^\circ$? Since $|OP| = |OQ| = |PQ|$ (all the same radius), triangle $OPQ$ is equilateral, so each angle is $60^\circ$. The angle sum of a triangle is $180^\circ$ [Theorem 4], and $180^\circ \div 3 = 60^\circ$.
Then bisect it with Construction 1 to get $30^\circ$. You can build other angles by combining these, for example $90^\circ$ (Construction 4) or $45^\circ$ (bisect $90^\circ$).
Practice★☆☆Not started
Using a compass and straight edge, construct an angle of $60^\circ$ and then bisect it. State the size of each resulting angle.
Using a compass and straight edge, construct an angle of $60^\circ$ and then bisect it. State the size of each resulting angle.
Construction 5 gives the line through a point $P$ that is parallel to a given line $\ell$. A neat way uses a rhombus (a four-sided shape with all sides equal). Opposite sides of a rhombus are parallel.
Practice★★☆Not started
Draw a line $\ell$ and mark a point $P$ not on $\ell$. Construct the line through $P$ parallel to $\ell$. Describe the steps you used.
Draw a line $\ell$ and mark a point $P$ not on $\ell$. Construct the line through $P$ parallel to $\ell$. Describe the steps you used.
Construction 6 divides a line segment into 2 or 3 equal parts using only a compass and straight edge. Construction 7 divides a segment into any number of equal parts (Higher Level only). Both use the fact that parallel lines cut transversals in equal ratios.
Practice★★☆Not started
Draw a line $\ell$ and mark a point $P$ not on $\ell$. Construct the line through $P$ parallel to $\ell$. Describe the steps you used.
Draw a line $\ell$ and mark a point $P$ not on $\ell$. Construct the line through $P$ parallel to $\ell$. Describe the steps you used.
A triangle is fixed (there is only one possible shape and size) when you know any of these:
Construction
Given
Also called
10
all three sides
SSS
11
two sides and the angle between them
SAS
12
two angles and a side
ASA
Construction 11 (SAS): draw one of the given sides, build the given angle at one end, and mark the second side along that arm using Construction 8. Then join the ends.
Practice★★☆Not started
Construct triangle $ABC$ with $AB = 6 \text{ cm}$, $BC = 5 \text{ cm}$ and $AC = 4 \text{ cm}$. Measure angle $B$ and state its size to the nearest degree.
Construct triangle $ABC$ with $AB = 6 \text{ cm}$, $BC = 5 \text{ cm}$ and $AC = 4 \text{ cm}$. Measure angle $B$ and state its size to the nearest degree.
Construction 15: a rectangle. The idea is to put perpendiculars at the ends of one side.
Back to the groundskeeper. She uses her rope as a compass. A rope pegged at each end of a line and swung in equal arcs gives the crossing point $Q$ of Construction 4. A third peg at $Q$ gives the square corner. Repeating this at the other end gives her rectangle.
Practice★★★Not started
Construct a right-angled triangle $ABC$ with hypotenuse $AC = 9 \text{ cm}$ and one other side $AB = 5 \text{ cm}$. a) Describe the steps of your construction. b) Use…
Construct a right-angled triangle $ABC$ with hypotenuse $AC = 9 \text{ cm}$ and one other side $AB = 5 \text{ cm}$.
a) Describe the steps of your construction.
b) Use Pythagoras' theorem to calculate $BC$ and check by measurement.
Construct a rectangle $ABCD$ with $AB = 8 \text{ cm}$ and $BC = 5 \text{ cm}$ using only a compass and straight edge. a) Describe your steps. b) Explain why the figure…
Construct a rectangle $ABCD$ with $AB = 8 \text{ cm}$ and $BC = 5 \text{ cm}$ using only a compass and straight edge.
a) Describe your steps.
b) Explain why the figure you drew is a rectangle, using properties of perpendicular and parallel lines.
Construct a triangle $ABC$ with $AB = 6 \text{ cm}$, angle $A = 45^\circ$ and $AC = 7 \text{ cm}$. Then construct the line through $C$ parallel to $AB$ and the line…
Construct a triangle $ABC$ with $AB = 6 \text{ cm}$, angle $A = 45^\circ$ and $AC = 7 \text{ cm}$. Then construct the line through $C$ parallel to $AB$ and the line through $B$ perpendicular to $AC$.
a) Describe your steps.
b) Measure the distance from $C$ to $AB$ and verify it using trigonometry.