Similar polygons: equal angles, proportional sides, and the similarity ratio k
Mathematics Prep 3 Term 1, Unit 3, Lesson 1
GoalDecide whether two polygons are similar, write the similarity statement in matching order, and find the ratio k.
New words
similar polygons · same shape, possibly different size: equal angles and proportional sidescorresponding · matching: the same position in the two polygonssimilarity ratio k · corresponding side of the first ÷ corresponding side of the secondABCD ~ EFGH · reads “ABCD is similar to EFGH”; A matches E, B matches F, and so on
Explanation
Two polygons are similar when they have the same shape, like a photo and its enlargement. Two things must both be true: (1) the corresponding angles are equal, and (2) the corresponding sides are proportional, that is all the ratios of corresponding sides are equal. This common ratio is the similarity ratio k = (side of the first) ÷ (matching side of the second). Going from the second polygon to the first, the ratio is 1 ÷ k. We write ABCD ~ EFGH with the vertices in matching order, so ∠A = ∠E, ∠B = ∠F, and AB ÷ EF = BC ÷ FG = CD ÷ GH = DA ÷ HE = k. One condition alone is not enough: a square and a rectangle have equal angles but not proportional sides, and a square and a rhombus have proportional sides but not equal angles. If k = 1 the polygons are congruent.
Worked examples
In the picture, ABCD and EFGH are parallelograms with all angles 60°, 120°, 60°, 120°. Prove ABCD ~ EFGH and find k. The angles match: A = E = 60°, B = F = 120°, C = G = 60°, D = H = 120°. The sides: AB ÷ EF = 12 ÷ 9 = 4/3 and BC ÷ FG = 8 ÷ 6 = 4/3; the other two sides repeat these (CD = AB, DA = BC and the same for EFGH). All ratios are equal, so ABCD ~ EFGH with k = 4/3. From EFGH to ABCD the ratio is 3/4.
Part
ABCD
EFGH
first side
AB = 10
EF = 15
second side
BC = 8
FG = 12
angle
∠A = 65°
∠E = 65°
angle
∠D = 90°
∠H = 90°
ABCD ~ EFGH, AB = 10 cm, EF = 15 cm, BC = 8 cm, ∠A = 65°, ∠B = 95°, ∠C = 110°. Find FG, ∠D and ∠H. k = AB ÷ EF = 10 ÷ 15 = 2/3. Then BC ÷ FG = 2/3, so FG = 8 × 3/2 = 12 cm. The angles of a quadrilateral add up to 360°: ∠D = 360° − 65° − 95° − 110° = 90°. ∠H corresponds to ∠D, so ∠H = 90°.
Are the rectangles 6 cm × 4 cm and 9 cm × 5 cm similar? All angles are 90°, so the angles agree. The sides: 9 ÷ 6 = 3/2 but 5 ÷ 4 = 5/4. The ratios differ, so the rectangles are not similar.
Class activity
Shape hunt. Look for two things at home that are similar (a photo and its enlargement, two tiles) and two that are not (a square and a rectangle). Measure two sides of each and test whether the ratios are equal.
Practice
1
The rectangles 15 cm × 10 cm and 6 cm × 4 cm: are they similar? Find k.
15 ÷ 6 = 5/2 and 10 ÷ 4 = 5/2. Angles are all 90°. Yes, similar, k = 5/2.
2
ABCD ~ EFGH with k = 3/2. If EF = 8 cm, find AB. What is the ratio from EFGH to ABCD?
AB = 8 × 3/2 = 12 cm. The reverse ratio is 2/3.
3
In ABCD ~ EFGH, ∠A = 70°, ∠B = 110° and ∠C = 105°. Find ∠H.
∠D = 360° − 70° − 110° − 105° = 75°, so ∠H = 75°.
4
A square and a rhombus with sides 4 cm and 6 cm and angles 60° and 120°: are the sides proportional? Are they similar?
All sides are equal in both, so the ratios agree (4 ÷ 6). But the angles are 90° and 60°/120°, so they are not similar.
5
If ABCD ~ EFGH, is EFGH ~ ABCD too? What is then k?
Yes. The angles still agree and the ratios are reversed: the new k is 1 ÷ k.
