Week 7 · The ratio of the areas of two similar triangles or polygons, scale drawings and maps · 3 lessons
2
Using the area ratio: the area cut off by a parallel line, the trapezium that remains and the perimeter from the ratio of areas
Mathematics Sec 1 Term 1, Unit 2, Lesson 3
GoalFind the area cut off by a line parallel to a side, the area of the remaining trapezium, and other lengths from a ratio of areas.
New words
trapezium · a quadrilateral with one pair of parallel sideswhole minus part · area of the trapezium = area of the big triangle − area of the small triangle
Explanation
A line DE parallel to BC cuts a small triangle ADE from △ABC, and △ADE ~ △ABC. The scale factor is k = AD/AB (the part to the whole, not AD/DB!). So area(ADE) = k² × area(ABC). The trapezium DBCE that is left is the whole minus the part: area(DBCE) = area(ABC) − area(ADE). To find k from a ratio like AD : DB = 3 : 2, add the parts: AB is 3 + 2 = 5 parts, so k = 3/5. Use the same ideas backwards: from two areas you get k, and then lengths or perimeters. A good habit is a quick check: the small triangle must have the smaller area, and the trapezium plus the triangle must give the whole.
Part
Area
△ADE (the small triangle)
k² × area(ABC)
trapezium DBCE
area(ABC) − area(ADE)
Worked examples
DE ∥ BC with AD : DB = 3 : 2 and area(ABC) = 200 m². AB = 5 parts, so k = 3/5 and area(ADE) = 200 × 9/25 = 72 m². The trapezium has 200 − 72 = 128 m².
AD = 3 × DB and area(ADE) = 72 cm². AB = 4 parts, so AD/AB = 3/4. area(ABC) = 72 ÷ 9/16 = 72 × 16/9 = 128 cm². The trapezium is 128 − 72 = 56 cm².
The sail of a felucca is a triangle of area 12 m². A similar sail twice as long in every side has area 12 × 2² = 48 m². At 15 LE per m² for the cloth, the cost goes from 180 LE to 720 LE: four times as much, not twice.
Class activity
Estimate first. A triangle has area 100 cm². A parallel line cuts off a triangle with k = 1/2. Before calculating, guess: is the cut-off part half of the area? Then calculate and compare.
Practice
1
DE ∥ BC with AD/AB = 2/3 and area(ABC) = 90 cm². Find area(ADE) and the trapezium.
