Similar triangles and area ratios · Grade 9
In Similar triangles and area ratios (Grade 9) you practise 4 skills: Matching sides and the AA test · The ratio of the areas · The ratio of the perimeters · Similar triangles in a figure. In similar triangles every side is multiplied by the same number, the scale factor. The perimeter is multiplied by the scale factor too, because it is the sum of the sides.
The area is multiplied by the square of the scale factor, because the base and the height are both multiplied by it. With a scale factor of 2, the area is 4 times larger. The other way round, take the square root of the ratio of the areas to find the scale factor. A line parallel to a side of a triangle makes a small triangle similar to the whole one. What is left is a trapezium: the area of the large triangle minus the area of the small one.
Tip: Do not mix them up: the perimeters are in the scale factor, and the areas are in its square.
What you learn
Matching sides and the AA test
For example: The triangles are similar: ΔABC ∼ ΔDEF . The lengths are in centimetres. In the first triangle, AB = 6, BC = 10, AC = 7 . In the second triangle, DF = 28 . What is the length of side EF ?
Answer and solution
The answer: 40 cm
Side DF matches side AC , so the scale factor is 28 ÷ 7 = 4 . The side you want matches side BC : EF = 10 × 4 = 40 . The length of the side is 40 cm.
The ratio of the areas
For example: Two similar triangles have a scale factor of 2. How many times larger is the area of the large triangle than the area of the small one?
Answer and solution
The answer: 4
The ratio of the areas is the square of the scale factor: 2² = 4 . The area is 4 times larger.
The ratio of the perimeters
For example: A large triangle is similar to a small one with a scale factor of 4. The perimeter of the small triangle is 10 cm. What is the perimeter of the large triangle?
Answer and solution
The answer: 40 cm
The perimeter of the large triangle: 10 × 4 = 40 . The perimeter is 40 cm.
Similar triangles in a figure
For example: In the picture, DE ∥ BC . In centimetres, AD = 6, DB = 6, BC = 6 . What is the length of DE ?
Answer and solution
The answer: 3 cm
The triangles are similar: ΔADE ∼ ΔABC . Side AB = 6 + 6 = 12 , and the scale factor: AD : AB = 6 : 12 = 1 : 2 . So DE = 6 ÷ 2 = 3 . The length is 3 cm.
Not on the site yet
Parts of this topic that are not built yet: Proving that triangles are similar · The SAS and SSS similarity tests.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education's grade 9 maths curriculum (2026–27).