7
Similar triangles and their properties
Textbook: pp. 30–31
GoalKnow the theorems on the ratio of perimeters, areas and heights of similar triangles and apply them.
New words
ratio of perimeters · perimetrlar nisbatiratio of areas · yuzlar nisbatik squared · k²corresponding heights · mos balandliklar
Explanation
The ratio of the perimeters of two similar triangles equals the similarity coefficient k, because every side is multiplied by k. The ratio of their areas is k²: S₁ : S = k². The ratio of corresponding heights (and medians, bisectors) is also k. The proof compares areas of triangles with a common angle through the products of their sides. For example, if k = 3 the perimeter becomes 3 times and the area 9 times as large. In the reverse problem we take the square root of the area ratio to find k.
Worked examples
A triangle has sides 5, 12, 13 cm (area 30 cm²). A similar triangle with k = 2 has perimeter 2 · 30 = 60 cm and area 4 · 30 = 120 cm².
Similar triangles have areas 18 cm² and 50 cm². The area ratio is 25/9, so k = 5/3. The side corresponding to a 6 cm side of the first is 6 · 5/3 = 10 cm.
Class activity
“k and k²”: each group fills in a table of the new perimeter and area for k = 2 and k = 3 for its own triangle, then finds the pattern from the table.
Practice
1
Two similar triangles have perimeters 24 cm and 36 cm. The smaller has area 20 cm². What is the area of the larger (cm²)? (k = 3/2)
45
2
Similar triangles have areas 16 cm² and 49 cm². Find the ratio of their sides.
4 : 7
3
If k = 5, how many times larger is the area?
25
4
Why is the ratio of areas k² and not k?
Area is two-dimensional: base and height each grow by k, so the area grows by k·k.