Exam practice

Every question from the end of every unit, with hints and answers to open. After you check an answer, mark how you did, and come back to the ones you didn’t get.

24 questions

Unit 15 · Construction

Exercise 15.1

Question 1

Construct a quadrilateral ABCD with AB = AD = 3 cm, BC = 2.5 cm, AC = 4 cm and BD = 5 cm.

Hint

Both diagonals are given. Draw AB; C comes from A and B (the diagonal AC and the side BC), and D from A and B too (the side AD and the diagonal BD).

Answer

Draw AB = 3 cm.
With center A and radius 4 cm, and center B and radius 2.5 cm, draw arcs that meet at C. Join BC (and the diagonal AC).
With center A and radius 3 cm, and center B and radius 5 cm, draw arcs that meet at D. Join AD, CD (and the diagonal BD): ABCD is the quadrilateral.

Figure 15.8

Check: CD ≈ 4.3 cm and ∠BAD ≈ 113°.

Question 2

Construct a quadrilateral MNOP with MO = MP = 6 cm, NO = 7.5 cm, OP = 5 cm and NP = 10 cm.

Hint

OP is a side and MO, NP are the diagonals. Draw OP first: M is 6 cm from both its ends, and N is 7.5 cm from O and 10 cm from P.

Answer

Draw OP = 5 cm.
With centers O and P and radius 6 cm, draw arcs that meet at M. Join PM (and the diagonal OM).
With center O and radius 7.5 cm, and center P and radius 10 cm, draw arcs that meet at N. Join NO, MN (and the diagonal NP): MNOP is the quadrilateral.

Figure 15.9

Check: the fourth side MN ≈ 4.7 cm.

Question 3

Construct a quadrilateral PQRS with PQ = 3.5 cm, QR = 2.5 cm, RS = 4 cm, ∠Q = 75° and ∠R = 120°.

Answer

Draw QR = 2.5 cm.
At Q construct ∠RQX = 75° and at R construct ∠QRY = 120°.
With center Q and radius 3.5 cm, cut QX at P; with center R and radius 4 cm, cut RY at S. Join PS: PQRS is the quadrilateral.

Figure 15.10

Check: PS ≈ 3.6 cm, ∠P ≈ 106° and ∠S ≈ 59° (the four angles add up to 360°).

Question 4

Construct a quadrilateral ABCD with AB = 3.6 cm, BC = 3.3 cm, AD = 2.7 cm and the diagonals AC = 4.6 cm and BD = 4 cm.

Answer

Draw AB = 3.6 cm.
With center A and radius 4.6 cm, and center B and radius 3.3 cm, draw arcs that meet at C. Join BC (and the diagonal AC).
With center A and radius 2.7 cm, and center B and radius 4 cm, draw arcs that meet at D. Join AD, CD (and the diagonal BD): ABCD is the quadrilateral.

Figure 15.11

Check: CD ≈ 2.7 cm.

Question 5

Construct a quadrilateral PQRS with PQ = 5 cm, QR = 6.5 cm, ∠P = ∠R = 100° and ∠S = 75°.

Hint

You know three angles, so you know the fourth. Which angle do you need at Q to place QR?

Answer

Draw PQ = 5 cm.
The angles add up to 360°, so ∠Q = 360° − 100° − 100° − 75° = 85°. With a protractor, draw ∠QPX = 100° at P and ∠PQY = 85° at Q.
With center Q and radius 6.5 cm, cut QY at R.
With a protractor, draw ∠QRZ = 100° at R. RZ meets PX at S: PQRS is the quadrilateral. Check that ∠S measures 75°.

Figure 15.12

100° and 85° can’t be made by bisecting, so they are drawn with a protractor. Check: RS ≈ 5.7 cm and PS ≈ 7.1 cm.

Question 6

Construct a quadrilateral ABCD with AB = 3.8 cm, BC = 3.4 cm, CD = 4.5 cm, AD = 5 cm and ∠B = 60°.

Hint

AB, BC and the angle between them make the triangle ABC. Then D is 5 cm from A and 4.5 cm from C.

Answer

Draw AB = 3.8 cm.
At B construct ∠ABX = 60°. With center B and radius 3.4 cm, cut BX at C.
With center A and radius 5 cm, and center C and radius 4.5 cm, draw arcs that meet at D. Join AD and CD: ABCD is the quadrilateral.

Figure 15.13

Check: the diagonal AC ≈ 3.6 cm and ∠A ≈ 115°.

Exercise 15.2

Question 1

Construct a trapezium ABCD with AB = 6 cm, BC = 4 cm, CD = 3.2 cm, ∠B = 75° and $DC \parallel AB$ .

Answer

Draw AB = 6 cm.
At B construct ∠ABX = 75°. With center B and radius 4 cm, cut BX at C.
At C construct ∠BCY = 105° (75° + 105° = 180°, so CY ∥ AB). With center C and radius 3.2 cm, cut CY at D. Join AD: ABCD is the trapezium.

Figure 15.22

Check: AD ≈ 4.2 cm and ∠A ≈ 65°.

Question 2

Construct a trapezium ABCD with $AB \parallel DC$ , AB = 7 cm, BC = 5 cm, AD = 6.5 cm and ∠B = 60°.

Hint

D is on the line through C parallel to AB, and 6.5 cm from A. Is it to the left of C or the right?

Answer

Draw AB = 7 cm.
At B construct ∠ABX = 60°. With center B and radius 5 cm, cut BX at C.
At C construct ∠BCY = 120°, so that CY ∥ AB.
With center A and radius 6.5 cm, cut CY at D. Join AD: ABCD is the trapezium.

Figure 15.23

C is only about 4.3 cm above AB, so the arc of 6.5 cm from A reaches the parallel line far to the left: D is about 4.8 cm to the left of A’s side, and DC ≈ 9.4 cm, longer than AB. That is fine: either parallel side may be the longer one. The same arc also meets the line just past C on the right, but that point makes AD cross BC, so it doesn’t give a quadrilateral. Check: ∠A ≈ 138° and ∠D ≈ 42°.

Question 3

Construct a trapezium ABCD with $AB \parallel CD$ , AB = 8 cm, BC = 6 cm, CD = 4 cm and ∠C = 120°.

Hint

∠B and ∠C are co-interior angles between the parallel sides.

Answer

Draw AB = 8 cm.
AB ∥ CD, so ∠B = 180° − 120° = 60° (co-interior angles). At B construct ∠ABX = 60°; with center B and radius 6 cm, cut BX at C.
At C construct ∠BCY = 120°. With center C and radius 4 cm, cut CY at D. Join AD: ABCD is the trapezium.

Figure 15.24

It is the trapezium of Example (a): ∠C = 120° says the same as ∠B = 60°. Check: AD ≈ 5.3 cm.

Question 4

Construct a trapezium PQRS with PQ = 10 cm, QR = 4 cm, RS = 6 cm, SP = 3 cm and $PQ \parallel SR$ .

Hint

As in Example (b): cut PT = SR = 6 cm on PQ. Then TR = PS = 3 cm and TQ = 4 cm.

Answer

Draw PQ = 10 cm.
With center P and radius 6 cm (= SR), cut PQ at T.
With center T and radius 3 cm (= SP), and center Q and radius 4 cm, draw arcs that meet at R. Join TR and QR.
With center R and radius 6 cm, and center P and radius 3 cm, draw arcs that meet at S. Join RS and PS: PQRS is the trapezium.

Figure 15.25

PTRS has PT = SR and TR = PS, so it is a parallelogram and $SR \parallel PQ$ . Check: ∠P ≈ 68° and ∠Q ≈ 44°.

Exercise 15.3

Question 1: from the two diagonals

Construct a rhombus PQRS with diagonals PR = 9.4 cm and QS = 7.2 cm.

Answer

Draw PR = 9.4 cm.
Draw the perpendicular bisector of PR; it meets PR at its midpoint O.
With center O and radius 3.6 cm (half of QS), cut the bisector above and below O at S and Q.
Join PQ, QR, RS and SP: PQRS is the rhombus.

Figure 15.34

Check: each side ≈ 5.9 cm (√(4.7² + 3.6²) = 5.92).

Construct a rhombus WXYZ with diagonals WY = 6.6 cm and XZ = 4.8 cm.

Answer

Draw WY = 6.6 cm.
Draw the perpendicular bisector of WY; it meets WY at its midpoint O.
With center O and radius 2.4 cm (half of XZ), cut the bisector above and below O at Z and X.
Join WX, XY, YZ and ZW: WXYZ is the rhombus.

Figure 15.35

Check: each side ≈ 4.1 cm.

Construct a rhombus ABCD with diagonals AC = 5.9 cm and BD = 6.8 cm.

Answer

Draw AC = 5.9 cm.
Draw the perpendicular bisector of AC; it meets AC at its midpoint O.
With center O and radius 3.4 cm (half of BD), cut the bisector above and below O at D and B.
Join AB, BC, CD and DA: ABCD is the rhombus.

Figure 15.36

Here BD is the longer diagonal, so the rhombus stands taller than it is wide. Check: each side ≈ 4.5 cm.

Question 2: from a side and an angle

Construct a rhombus EFGH with side EF = 6.3 cm and ∠E = 75°.

Answer

Draw EF = 6.3 cm.
At E construct ∠FEH = 75°, and with center E and radius 6.3 cm, cut its arm at H.
With centers H and F and the same radius 6.3 cm, draw arcs that meet at G. Join the sides.

Figure 15.37

Check: ∠F = 105°; the diagonals ≈ 10.0 cm and 7.7 cm.

Construct a rhombus ABCD with side BC = 5.5 cm and ∠C = 60°.

Answer

Draw BC = 5.5 cm.
At C construct ∠BCD = 60°, and with center C and radius 5.5 cm, cut its arm at D.
With centers D and B and the same radius 5.5 cm, draw arcs that meet at A. Join the sides.

Figure 15.38

Check: the diagonal BD = 5.5 cm (the triangle BCD is equilateral) and AC ≈ 9.5 cm.

Construct a rhombus IJKL with side IJ = 5.1 cm and ∠I = 105°.

Answer

Draw IJ = 5.1 cm.
At I construct ∠JIL = 105°, and with center I and radius 5.1 cm, cut its arm at L.
With centers L and J and the same radius 5.1 cm, draw arcs that meet at K. Join the sides.

Figure 15.39

105° is made by bisecting the angle between 90° and 120°. Check: ∠J = 75°; the diagonals ≈ 8.1 cm and 6.2 cm.

Construct a rhombus WXYZ with side WX = 6.2 cm and ∠Y = 45°.

Hint

∠Y is not at either end of WX. Which angle is equal to it?

Answer

Opposite angles of a rhombus are equal, so ∠W = ∠Y = 45°. Draw WX = 6.2 cm.
At W construct ∠XWZ = 45°, and with center W and radius 6.2 cm, cut its arm at Z.
With centers Z and X and the same radius 6.2 cm, draw arcs that meet at Y. Join the sides.

Figure 15.40

Check: ∠X = ∠Z = 135°; the diagonals ≈ 11.5 cm and 4.7 cm.

Question 3: from a side and a diagonal

Construct a rhombus ABCD with side AB = 5.8 cm and the diagonal BD = 7.3 cm.

Answer

Draw BD = 7.3 cm.
Draw the perpendicular bisector of BD; it meets BD at its midpoint O.
With center B and radius 5.8 cm, cut the bisector above and below BD at A and C.
Join BC, CD, DA and AB: ABCD is the rhombus.

Figure 15.41

Check: the other diagonal AC ≈ 9.0 cm.

Construct a rhombus PQRS with side QR = 4.3 cm and the diagonal PR = 6.5 cm.

Answer

Draw PR = 6.5 cm.
Draw the perpendicular bisector of PR; it meets PR at its midpoint O.
With center P and radius 4.3 cm, cut the bisector above and below PR at S and Q.
Join PQ, QR, RS and SP: PQRS is the rhombus.

Figure 15.42

All four sides are equal, so the arcs of 4.3 cm may be drawn from P or from R. Check: QS ≈ 5.6 cm.

Construct a rhombus EFGH with side EF = 5.1 cm and the diagonal EG = 8.2 cm.

Answer

Draw EG = 8.2 cm.
Draw the perpendicular bisector of EG; it meets EG at its midpoint O.
With center E and radius 5.1 cm, cut the bisector above and below EG at H and F.
Join EF, FG, GH and HE: EFGH is the rhombus.

Figure 15.43

Check: FH ≈ 6.1 cm.

Question 4: from a side and the angle a diagonal makes with it

Construct a rhombus ABCD with side AB = 5.3 cm and ∠ABD = 30°.

Hint

BD is a diagonal. Make the 30° at B; then D is on its arm, 5.3 cm from A.

Answer

Draw AB = 5.3 cm.
At B construct ∠ABX = 30°.
With center A and radius 5.3 cm, cut BX at D (so BD is a diagonal).
With centers B and D and the same radius 5.3 cm, draw arcs that meet at C. Join the sides.

Figure 15.44

The diagonal bisects ∠B, so ∠ABC = 60° and ∠A = 120°. Check: BD ≈ 9.2 cm.

Construct a rhombus PQRS with side QR = 6.1 cm and ∠PQR = 75°.

Answer

Draw QR = 6.1 cm.
At Q construct ∠RQP = 75°, and with center Q and radius 6.1 cm, cut its arm at P.
With centers P and R and the same radius 6.1 cm, draw arcs that meet at S. Join the sides.

Figure 15.45

∠PQR is the angle between two sides, so this is a construction of the kind in (b). Check: ∠P = 105°; the diagonals ≈ 9.7 cm and 7.4 cm.

Construct a rhombus EFGH with side GH = 4.6 cm and ∠GEF = 60°.

Answer

Draw EF = 4.6 cm.
At E construct ∠FEX = 60°.
With center F and radius 4.6 cm, cut EX at G (so EG is a diagonal).
With centers E and G and the same radius 4.6 cm, draw arcs that meet at H. Join the sides.

Figure 15.46

All sides are 4.6 cm, so EF = 4.6 cm. As in (d), the triangle EFG is equilateral: EG = 4.6 cm and ∠FEH = 120°.

Construct a rhombus WXYZ with side WX = 5.6 cm and ∠WXZ = 45°.

Answer

Draw WX = 5.6 cm.
At X construct ∠WXT = 45°.
With center W and radius 5.6 cm, cut XT at Z (so XZ is a diagonal).
With centers X and Z and the same radius 5.6 cm, draw arcs that meet at Y. Join the sides.

Figure 15.47

The diagonal XZ bisects ∠X, so ∠WXY = 90°: this rhombus is a square, with every angle 90° and both diagonals ≈ 7.9 cm (5.6 × √2).