Let us memorize the side ratios of two commonly used types of right triangles.
45°-45°-90° Isosceles Right Triangle
The ratio of the three sides is 1:1:√2. If one side is a, the hypotenuse is a√2.
30°-60°-90° Right Triangle
The ratio of the three sides is 1:√3:2. If the shortest side is a, the middle-length side is a√3 and the hypotenuse is 2a.
📘 Example ①
Find the length of the diagonal of a square with side length 6cm.
Answer: The diagonal is the hypotenuse of a 45-45-90 triangle. 6√2 = 6√2 cm
Find the length of the diagonal of a square with side length 6cm.
Answer: The diagonal is the hypotenuse of a 45-45-90 triangle. 6√2 = 6√2 cm
📘 Example ②
Find the height of an equilateral triangle with side length 6cm.
Answer: In a 30-60-90 triangle, the shortest side = 3cm. Height = 3√3 = 3√3 cm
Find the height of an equilateral triangle with side length 6cm.
Answer: In a 30-60-90 triangle, the shortest side = 3cm. Height = 3√3 = 3√3 cm
💡 Key Point
- Isosceles right triangle: 1:1:√2
- 30-60-90 triangle: 1:√3:2
Practice Problems
- Find the length of the diagonal of a square with side length 4cm.
- Find the height of an equilateral triangle with side length 4cm.
Answers and Explanations
- Answer: 4√2 cm.
- Answer: 2√3 cm. Height = 4 × √3/2 = 2√3.