8 min readPathway sheetPart IV construction (8 marks), either-or choice

Construction: a similar triangle with a given scale factor

Mark equal steps on a ray, join, then draw parallel lines. The new triangle is similar, with each side scale factor × old side.

Scale factor m/nNew side = (m/n) × old side; less than 1 shrinks, more than 1 enlarges

Joining ruleMark the larger of m and n points; join the nth point; draw a parallel through the mth point

Why it worksThales theorem and the AA criterion

InstrumentsRuler, compasses, sharp pencil, set squares (protractor if an angle is given)

What a scale factor means

A scale factor tells how many times bigger or smaller the new triangle is. Scale factor 3/5 makes each new side 3/5 of the old side, so the triangle shrinks. Scale factor 7/4 makes each new side 7/4 of the old side, so it enlarges. The angles do not change. So the new triangle is similar to the old one.

Steps of construction: scale factor 3/5 (smaller triangle)

Question: ΔPQR has QR = 6 cm, PQ = 5 cm and PR = 4.5 cm. Construct a similar triangle with sides 3/5 of those of ΔPQR. Instruments: ruler, compasses, sharp pencil, set squares.

1
Draw a rough figure first. Show ΔPQR, a ray QX below QR with points Q₁ to Q₅, and the small triangle P′QR′. Write the given lengths on it.
2
Draw QR = 6 cm. With Q as centre and radius 5 cm, draw an arc. With R as centre and radius 4.5 cm, cut it at P. Join PQ and PR.
3
Draw a ray QX from Q, making an acute angle with QR, on the side opposite to P.
4
With compasses, mark 5 equal steps on QX: Q₁, Q₂, Q₃, Q₄, Q₅.
use 5, the larger of 3 and 5
5
Join Q₅R.
5 is the denominator
6
Through Q₃, draw a line parallel to Q₅R. It meets QR at R′.
3 is the numerator
7
Through R′, draw a line parallel to RP. It meets QP at P′.
8
ΔP′QR′ is the required triangle. Each side is 3/5 of the matching side of ΔPQR.

Steps of construction: scale factor 7/4 (larger triangle)

Question: ΔPQR has QR = 4 cm, PQ = 3.6 cm and PR = 4.4 cm. Construct a similar triangle with sides 7/4 of those of ΔPQR. Instruments: ruler, compasses, sharp pencil, set squares.

1
Draw a rough figure. Show ΔPQR, a ray QX below QR with points Q₁ to Q₇, and QR and QP extended to R′ and P′. Write the given lengths.
2
Draw QR = 4 cm. Cut arcs of 3.6 cm from Q and 4.4 cm from R. Mark P where they meet. Join PQ and PR.
3
Draw a ray QX from Q, making an acute angle with QR, on the side opposite to P.
4
Mark 7 equal steps on QX: Q₁ to Q₇.
use 7, the larger of 7 and 4
5
Join Q₄R.
4 is the denominator
6
Through Q₇, draw a line parallel to Q₄R. It meets QR extended at R′.
7
Through R′, draw a line parallel to RP. It meets QP extended at P′.
8
ΔP′QR′ is the required triangle. Each side is 7/4 of the matching side of ΔPQR.

Example: work out the lengths to check your drawing

Problem

Before you write 'required triangle', find the sides of ΔP′QR′ in both constructions above. Then measure your drawing.

Working

Scale factor 3/5: QR′ = (3/5) × 6 = 3.6 cm
QP′ = (3/5) × 5 = 3 cm and P′R′ = (3/5) × 4.5 = 2.7 cm
Scale factor 7/4: QR′ = (7/4) × 4 = 7 cm
QP′ = (7/4) × 3.6 = 6.3 cm and P′R′ = (7/4) × 4.4 = 7.7 cm
Measure your triangle. A difference of about 1 mm is normal.

Answer

Scale factor 3/5: 3.6 cm, 3 cm, 2.7 cm. Scale factor 7/4: 7 cm, 6.3 cm, 7.7 cm.

Why it works

Q₃R′ is parallel to Q₅R. By Thales theorem, QR′/QR = QQ₃/QQ₅ = 3/5. Next, R′P′ is parallel to RP, so ∠QR′P′ = ∠QRP (corresponding angles). ∠Q is common to both triangles. So ΔP′QR′ ~ ΔPQR by AA, and every side is in the ratio 3/5.

In real life

A photocopier can enlarge or reduce a page by a scale factor, such as 70% or 141%. Maps, model buildings and toy cars are similar to the real things by a scale factor.

Exam angle

1 marks: “To construct a triangle with scale factor 5/3, how many equal points do you mark on the ray?” 5, the larger of 5 and 3.

2 marks: “A triangle has sides 6 cm, 8 cm and 10 cm. It is reduced to sides 4.5 cm, 6 cm and 7.5 cm. Find the scale factor.” 4.5/6 = 6/8 = 7.5/10 = 3/4. The scale factor is 3/4, which is less than 1, so it is a reduction.

8 marks: “ΔXYZ has YZ = 5 cm, XY = 4 cm and XZ = 6 cm. Construct a similar triangle with sides 4/3 of the corresponding sides of ΔXYZ.” Rough figure; draw ΔXYZ; ray YW below YZ with Y₁ to Y₄; join Y₃Z. Through Y₄ draw a parallel to Y₃Z, meeting YZ extended at Z′. Through Z′ draw a parallel to ZX, meeting YX extended at X′. Check: YZ′ ≈ 6.7 cm, YX′ ≈ 5.3 cm, X′Z′ = 8 cm.

Remember it

Join the bottom, parallel through the top: join the denominator point to the vertex, then draw the parallel through the numerator point.