Secondary 2 Geometric Constructions

All 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Constructing Perpendicular and Parallel lines

   Draw a line $d$ right across your page, and mark a point $P$ about $4\text{cm}$ above it. Use only a compass and an unmarked straightedge — no protractor, no ruler markings.

   1. Construct the line through $P$ that is perpendicular to $d$. Call it $p$.
   2. Construct the line through $P$ that is parallel to $d$.
   3. Explain what guarantees that the line you drew in b) never meets $d$.
2. Q2Constructing Polygons

   Using a ruler and a protractor, construct the quadrilateral $MATH$ with $$MA=6\text{cm},\angle A={110}^{\circ },AT=4.5\text{cm},\angle T={75}^{\circ },TH=5\text{cm}.$$ Build it in that order, then join $H$ back to $M$.

   1. Measure $\angle M$ and $\angle H$ on your drawing.
   2. Without measuring anything else, say what $\angle M+\angle H$ must be, and use it to check your two measurements.
3. Q3Constructing Regular Polygons

   Open your compass to $4\text{cm}$ and draw a circle. Without changing the opening, put the compass point anywhere on the circle and mark an arc on the circle; move the point to that new mark and repeat, all the way round.

   1. Join the marks in order. Which regular polygon have you built? Measure a side to check.
   2. Draw the segments from the centre to each vertex. What kind of triangle is each of the six pieces, and why?
   3. Use b) to explain why the marks land exactly back on the starting point, instead of overshooting or falling short.
4. Q4Constructing a Bisector

   Draw an angle $\angle BAC$ of ${76}^{\circ }$ with your protractor, making both arms about $7\text{cm}$ long.

   1. Construct its bisector with a compass and straightedge only.
   2. Measure the two halves with your protractor. What should each one read?
   3. Mark a point $P$ on the bisector, about $3\text{cm}$ from $A$. Measure the shortest distance from $P$ to each arm. What do you notice, and why does the construction force it?
5. Q5Constructing a Circle

   Construct a circle with centre $O$ and diameter $7\text{cm}$.

   1. What compass opening did you use, and why?
   2. Draw a chord $AB$ of length $6\text{cm}$ in your circle (not through the centre).
   3. Construct the perpendicular from $O$ to the chord $AB$, and call its foot $M$. Measure $AM$ and $MB$. State the property your measurements suggest.
6. Q6Constructing a Median

   Construct a triangle $ABC$ with $AB=9\text{cm}$, $BC=7\text{cm}$ and $AC=6\text{cm}$.

   1. Construct the midpoint of each side, and draw the three medians.
   2. What do you notice about the three medians? Call the point $G$.
   3. Measure $AG$ and the distance from $G$ to the midpoint of $BC$. What relationship do you see between the two?
7. Q7Constructing a Perpendicular Bisector

   Draw a segment $AB$ of length $8\text{cm}$.

   1. Construct its perpendicular bisector with a compass and straightedge.
   2. Mark two points $P$ and $Q$ on that line, one on each side of $AB$. Measure $PA$, $PB$, $QA$ and $QB$, then state the property these four measurements show.
   3. On a map, two skateparks stand $8\text{cm}$ apart. The city wants a water fountain the same distance from both. Where can it be placed? How many possible positions are there?
8. Q8Constructing a Square and a Rectangle

   Construct a rectangle $ABCD$ with $AB=6\text{cm}$ and $BC=4\text{cm}$, using a compass and straightedge for the right angles rather than the corner of your ruler.

   1. Describe the order of your steps.
   2. Draw both diagonals and measure them. What do you notice?
   3. Mark the point where the diagonals cross and measure the four pieces. State the two properties of a rectangle's diagonals that your measurements support.
9. Q9Constructing a Trapezoid

   Construct a trapezoid $PQRS$ whose parallel sides measure $PQ=7\text{cm}$ and $SR=4\text{cm}$, with a height of $3\text{cm}$ between them. Place $S$ so that it sits $1.5\text{cm}$ to the right of $P$.

   1. Describe how you used a perpendicular and a parallel construction to place the shorter side.
   2. Calculate the area of your trapezoid.
   3. Amélie builds a different trapezoid with the same two parallel sides and the same height, but with $S$ placed $3\text{cm}$ to the right of $P$. Without measuring, say how its area compares with yours, and why.
10. Q10Constructing a Triangle

    Build each triangle accurately, then answer the question that follows it.

    1. Construct $△ABC$ with $AB=7\text{cm}$, $BC=6\text{cm}$ and $AC=4\text{cm}$, using a compass. Measure its largest angle. Which side is it opposite?
    2. Construct $△DEF$ with $DE=8\text{cm}$, $\angle D={40}^{\circ }$ and $\angle E={65}^{\circ }$. Find $\angle F$ without measuring it, then check with your protractor.
11. Q11The Construction of a Height

    Construct $△ABC$ with $BC=6\text{cm}$, $\angle B={130}^{\circ }$ and $AB=4\text{cm}$.

    1. Construct the height from $A$ to the side $BC$. Something unusual happens — describe it, and say what you must do to the side $BC$ before the construction will work.
    2. Measure that height and calculate the area of the triangle.
    3. Construct the height from $B$ to the side $AC$. Does it behave the same way? Explain the difference in terms of the angles of the triangle.
12. Q12Synthesis — drawing on several sheets in this topic

    Construct $△ABC$ with $AB=8\text{cm}$, $BC=7\text{cm}$ and $AC=5\text{cm}$.

    1. Construct the perpendicular bisector of $AB$ and the perpendicular bisector of $BC$. Call their crossing point $O$.
    2. Draw the circle with centre $O$ and radius $OA$. What do you notice about $B$ and $C$?
    3. Explain, using the property of a perpendicular bisector, why this had to happen. What would the perpendicular bisector of the third side, $AC$, do if you constructed it too?

The 7 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The drawing is the easy half

A construction question at this level has two parts, and only the first one is done with a compass. Q1 asks for a perpendicular and a parallel, and then asks what guarantees that the parallel never meets the original line. Q4 asks for the bisector of an angle, and then for the property that forces a point on it to be the same distance from both arms. Q5 asks you to drop a perpendicular from a centre to a chord, and then to state what your two measurements suggest. In every case the drawing is the evidence and the sentence afterwards is the answer.

That structure is deliberate. A construction that lands correctly by luck looks exactly like one that lands correctly by method, and only the justification tells them apart. Write the sentence even where the question does not carve out a part for it.

Leave the arcs on the page

The single most common way to lose marks here is to rub out the construction arcs once the line is drawn. Those arcs are the method: they show that the perpendicular came from two equal radii rather than from the edge of a ruler, and without them the drawing claims nothing. Draw them lightly, draw them long enough to cross properly, and leave them.

Which tools a question allows is part of the question (Q1, Q2, Q8)

Q1 says compass and unmarked straightedge only — no protractor, no ruler markings — so a right angle has to be constructed rather than measured. Q2 hands you a ruler and a protractor and asks for a quadrilateral from given sides and angles, which is a different skill: measuring accurately and building in the stated order. Q8 sits in between, asking for a rectangle whose right angles come from a compass construction rather than from the corner of the ruler.

Read that instruction before starting, because a correct figure built with the wrong tools does not answer the question that was asked. It is also the reason the constructions are worth learning at all: they produce exact right angles, exact midpoints and exact bisections from nothing but equal distances, and no measuring instrument is exact.

Build in an order, and say what the order was (Q2, Q8, Q9)

Q8 and Q9 both ask you to describe your steps, which is the written half of a construction. A useful discipline: start from the longest given segment, add the angles or perpendiculars that hang off it, and only then place the remaining vertices. In Q9 the shorter parallel side is placed by first raising a perpendicular of the right height and then drawing a parallel through its top — the height and the parallel are two separate constructions, and describing them separately is what the question wants.

Q2 ends with a check that costs nothing: the four interior angles of a quadrilateral add to 360°, so the two angles you built and the two you measured must complete that total. If they do not, the drawing has drifted, and knowing that before you answer anything else is worth more than the measurement itself.

The properties the drawings are there to reveal (Q5, Q6, Q7, Q12)

Several questions are constructions in form and discoveries in purpose. The measurement at the end is not busywork — it is the evidence for a statement you are then asked to make.

When the height falls outside the triangle (Q11)

Q11 hands you an obtuse triangle and asks for a height, and the construction fails as stated: the perpendicular from the vertex does not meet the opposite side anywhere. The resolution is to extend that side beyond the vertex and drop the perpendicular onto the extension. The height is the segment from the vertex to that foot, and it is still the correct length to use in the area formula, even though part of it lies outside the triangle.

The question then asks whether the other height behaves the same way, and it does not — which is the useful observation. In an obtuse triangle two of the three heights fall outside; in a right triangle two of them are the sides themselves; in an acute triangle all three fall inside. The obtuse angle is what decides it.

Nets, and why an inexact triangle will not close

A net is a construction problem with a physical test attached: fold it up and either the edges meet or they do not. Two lengths that were supposed to be equal but were cut freehand leave a gap that no amount of care in the folding recovers, which is exactly why the triangles of a net are built with a compass — equal radii give equal sides — and why the fold lines are drawn as perpendiculars rather than estimated. If you build one, count its faces, edges and vertices before folding: it is a check on the net itself, and it costs nothing.

Getting the most out of it

Bring the tools, and give the drawing room

A compass with a firm hinge, a sharp pencil, a straightedge, a ruler and a protractor. Work at the sizes given rather than scaling down to fit a corner of the page — the properties these questions are looking for are visible at 8 cm and invisible at 3 cm.

Never erase the construction arcs

They are the proof that the line came from a construction rather than from a guess. A finished answer shows the arcs, the crossing points they create, and the final line drawn firmly through them.

Finish every drawing with a sentence

"Every point on this line is the same distance from A and from B." "The two perpendiculars cannot meet, so the lines are parallel." That sentence is the part being assessed, and it is also the part that is still worth something if the drawing came out slightly off.

Measure to test, not to build

Where a question allows only a compass and a straightedge, the protractor is for checking afterwards. If your constructed bisector reads 38° on each side of a 76° angle, the construction worked — and you have verified it rather than assumed it.

Common questions