Review: similarity of polygons and triangles, areas, chords and tangents, proportional parts and angle bisectors
Mathematics Sec 1 Term 1, Units 1–4, Review
GoalRevise Unit 2 with new mixed problems.
New words
scale factor · k: the ratio of corresponding lengthssecant · a line cutting a circle at two pointsproportional parts · segments cut by parallels in equal ratios
Explanation
Checklist for Unit 2. (1) Similar figures: equal angles and sides in the ratio k; the ratio of perimeters is k and the ratio of areas is k². (2) AA is enough for triangles. (3) Two chords through E: EA × EB = EC × ED. Two secants from an outside point E: EA × EB = EC × ED. A tangent EA and a secant: EA² = EC × ED. (4) A line parallel to BC cutting AB at D and AC at E gives AD/DB = AE/EC, and DE/BC = AD/AB. (5) The interior bisector from A divides BC in the ratio AB : AC, the exterior one in the same ratio externally; AD² = AB × AC − BD × DC. (6) For map scales, multiply lengths by the scale and areas by the scale squared.
Worked examples
Two similar polygons have scale factor 3/2. A side of 8 cm becomes 12 cm, a perimeter of 30 cm becomes 45 cm and an area of 40 cm² becomes 40 × 9/4 = 90 cm².
On a map with scale 1 : 50 000 a road is 4.2 cm long, so it is 4.2 × 50 000 = 210 000 cm = 2100 m = 2.1 km. A map square of 1 cm² stands for 500 m × 500 m = 250 000 m² = 25 hectares.
Chords AB and CD meet at E inside a circle with EA = 4, EB = 9, EC = 6. EA × EB = EC × ED gives 36 = 6x, so ED = 6.
From E a tangent EA and a secant through C and D with EC = 4, CD = 5. ED = 9, so EA² = EC × ED = 36 and EA = 6.
DE ∥ BC with D on AB and E on AC, AD = 6, DB = 4, AE = 9. 6/4 = 9/EC gives EC = 6; DE/BC = 6/10 so if BC = 20, DE = 12. Bisector: AB = 15, AC = 9, BC = 16 gives BD/DC = 5/3, BD = 10, DC = 6 and AD² = 135 − 60 = 75.
Class activity
Mini survey. Draw a plan of your classroom to the scale 1 : 50 (2 cm for 1 m). Measure the length and width of the floor, find the real area in m² and the area on the plan in cm², and check that the ratio of areas is 2500.
Practice
1
Two similar triangles have areas 54 cm² and 24 cm². Find the ratio of their sides.
Area ratio 9/4, so k = 3/2.
2
Chords AB and CD meet at E: EA = 3, EB = 8, EC = 4. Find ED.
24 = 4 × ED, ED = 6.
3
A tangent from E has length 10 and the secant has outer part 4. Find the whole secant.
100 = 4 × whole, whole = 25.
4
AD bisects ∠A: AB = 12, AC = 20, BC = 24. Find BD and DC.
Ratio 3/5, 8 parts for 24: BD = 9, DC = 15.
5
A field on a map of scale 1 : 20 000 has area 6 cm². Find its real area in hectares.
1 cm = 200 m, so 1 cm² = 40 000 m² = 4 ha; 6 cm² = 24 ha.
