Similar special shapes: squares, rectangles, rhombuses, parallelograms and regular polygons
Mathematics Prep 3 Term 1, Unit 3, Lesson 1
GoalUse the short tests for similarity of squares, rectangles, rhombuses, parallelograms and regular polygons.
New words
regular polygon · a polygon with all sides equal and all angles equalrhombus · a quadrilateral with four equal sidesparallelogram · a quadrilateral with two pairs of parallel sidesproportional · with equal ratios
Explanation
For some shapes one short test is enough. All squares are similar (all angles 90°, all sides equal). Two regular polygons with the same number of sides are similar (their angles are equal and all their sides are equal); a regular pentagon and a regular hexagon are not, because their angles differ (108° and 120°). Two rectangles are similar when their corresponding dimensions are proportional: length ÷ length = width ÷ width. Two rhombuses are similar when one pair of angles is equal (a rhombus has equal sides, so the side ratios are automatically equal, and the other angles follow). Two parallelograms are similar when one pair of angles is equal and the sides around those angles are proportional. Remember that in a rhombus with a 40° angle the other angles are 140°, 40°, 140°.
Shapes
When are two of them similar?
squares
always
regular polygons
same number of sides
rectangles
length ÷ length = width ÷ width
rhombuses
one pair of equal angles
parallelograms
one pair of equal angles and proportional sides around them
Worked examples
Are the rectangles 12 × 8 and 9 × 6 similar? And 12 × 8 and 15 × 9? 12 ÷ 9 = 4/3 and 8 ÷ 6 = 4/3: similar, k = 4/3. 12 ÷ 15 = 4/5 but 8 ÷ 9 is not 4/5: not similar.
Rhombus ABCD has side 6 cm and ∠A = 50°. Rhombus EFGH has side 9 cm and ∠E = 50°. Are they similar? Yes: one pair of equal angles is enough for rhombuses, and k = 6 ÷ 9 = 2/3. A rhombus with ∠E = 60° would not be similar to ABCD, because its angles 60° and 120° differ from 50° and 130°.
Two regular hexagons have sides 4 cm and 10 cm. Are they similar, and what is k? Every angle of a regular hexagon is (6 − 2) × 180° ÷ 6 = 120°, and all sides are equal, so yes. k = 4 ÷ 10 = 2/5. A regular pentagon (angles 108°) is not similar to a regular hexagon.
Class activity
Rectangle test. Cut a rectangle from paper, 12 cm by 8 cm. Cut another with sides 9 cm by 6 cm and another 15 cm by 9 cm. Lay them corner to corner: for a similar pair the far corners lie on a straight line (the diagonal). Which pair lines up?
Practice
1
Are the parallelograms ABCD and EFGH similar? ∠A = ∠E = 50°, AB = 10, AD = 6, EF = 15, EH = 9.
10 ÷ 15 = 2/3 and 6 ÷ 9 = 2/3, around the equal angles. Yes, k = 2/3.
2
The same parallelogram ABCD and a parallelogram with ∠E = 50°, EF = 15, EH = 8. Similar?
10 ÷ 15 = 2/3 but 6 ÷ 8 = 3/4. The ratios differ, so they are not similar.
3
A rhombus has an angle of 70°, another has an angle of 110°. Are they similar?
Yes. Each rhombus has angles 70° and 110°, so one pair of angles is equal.
4
Are all rectangles similar? Give an example.
No. A 2 × 1 rectangle and a 2 × 2 square have equal angles but 2 ÷ 2 = 1 and 1 ÷ 2 = 1/2.
5
Are a regular pentagon and a regular octagon similar? Give the reason with angles.
No. A regular pentagon has angles (5 − 2) × 180° ÷ 5 = 108°, a regular octagon (8 − 2) × 180° ÷ 8 = 135°.
