Week 5 · Similarity of polygons, the scale factor and the AA postulate for triangles · 3 lessons
1
Similar polygons: equal corresponding angles and proportional corresponding sides, the scale factor k and the order of the letters
Mathematics Sec 1 Term 1, Unit 2, Lessons 1–2
GoalDecide whether two polygons are similar and find missing sides with the scale factor k.
New words
similar polygons · same shape, different size: equal corresponding angles and proportional corresponding sidesscale factor k · the ratio of a side of the first polygon to the matching side of the secondcorresponding · matching: the sides or angles that sit in the same place in the two figures
Explanation
Two polygons are similar when they have the same number of sides, all their corresponding angles are equal AND their corresponding sides are proportional. Both conditions are needed: a square and a rectangle have equal angles, but their sides are not proportional; a square and a rhombus have proportional sides, but their angles differ. We write ABCD ~ EFGH, and the order of the letters shows the correspondence: ∠A = ∠E, ∠B = ∠F, ∠C = ∠G, ∠D = ∠H, and AB/EF = BC/FG = CD/GH = DA/HE = k. This common ratio k is the scale factor. In this course k always runs from the first polygon to the second: k > 1 means the first is larger, 0 < k < 1 the first is smaller, k = 1 means congruent. All squares are similar, all circles are similar, all equilateral triangles are similar, but not all rectangles are.
Pair of shapes
Angles equal?
Sides proportional?
Similar?
two squares
yes
yes
yes
square and rectangle
yes
no
no
square and rhombus
no
yes
no
Worked examples
ABCD ~ EFGH with AB = 28, BC = 24, CD = 18, DA = 26 and EF = 14, FG = 12. Find GH and HE. The scale factor is k = AB/EF = 28/14 = 2 (check BC/FG = 24/12 = 2 ✓). Then GH = CD/k = 18/2 = 9 and HE = DA/k = 26/2 = 13.
Are a 6 × 4 rectangle and a 9 × 6 rectangle similar? All angles are 90°. The ratios of the sides are 6/9 = 2/3 and 4/6 = 2/3, equal, so they are similar with k = 2/3. A 6 × 4 and an 8 × 6 rectangle are not similar: 6/8 = 3/4 but 4/6 = 2/3.
△ABC ~ △DEF with AB = 6, BC = 8, CA = 10 and DE = 9. Then k = 6/9 = 2/3, so EF = BC/k = 8 × 3/2 = 12 and FD = CA/k = 10 × 3/2 = 15.
Class activity
Photocopy test. Draw a rectangle 6 cm by 4 cm. Imagine copying it so that the width becomes 9 cm. What must the new height be to keep the same shape? What is k?
Practice
1
ABCD ~ EFGH with AB = 10, EF = 4 and BC = 15. Find FG.
k = 10/4 = 5/2, so FG = 15 ÷ 5/2 = 6.
2
ABCD ~ EFGH and m(∠A) = 70°. What is m(∠E)? Why?
70°, because A and E are corresponding vertices in the order of the letters.
3
△ABC ~ △DEF with AB = 4, BC = 7, CA = 9 and DE = 6. Find EF and FD.
k = 4/6 = 2/3; EF = 7 × 3/2 = 10.5 and FD = 9 × 3/2 = 13.5.
4
Are a 5 × 3 rectangle and a 20 × 12 rectangle similar? Give k.
5/20 = 1/4 and 3/12 = 1/4: yes, similar with k = 1/4.
5
Is every rhombus similar to every other rhombus? Explain with one example.
No. A square is a rhombus, and so is a thin rhombus with angles 60° and 120°; their sides are proportional but their angles are different.
