Foundation Teaching Library
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Quantitative Articles (2)
Quantitative•6 Min Read
Special Right Triangles & Pythagorean Theorems
Master the fundamental side ratios of 45°-45°-90° and 30°-60°-90° triangles to solve spatial Qiyas problems in under 30 seconds.
1. The Isosceles Right Triangle (45°-45°-90°)
When a right triangle has two equal interior angles of 45°, it forms an isosceles right triangle. The legs opposite the 45° angles are equal in length. The hypotenuse opposite the 90° angle is always equal to the leg length multiplied by the square root of 2.
Core Takeaway
Shortcut Rule: If the leg is 5, the hypotenuse is 5√2 without needing full Pythagorean calculation.
2. The 30°-60°-90° Special Right Triangle
In a 30°-60°-90° right triangle, the side opposite the 30° angle is the shortest leg. The hypotenuse is exactly twice the length of the shortest leg. The longer leg opposite the 60° angle is equal to the shortest leg multiplied by the square root of 3.
Core Takeaway
Always locate the 30° angle first. The shortest side opposite 30° serves as the base multiplier for all other sides.
Self-Assessment Check
In a 30°-60°-90° right triangle, if the shortest leg opposite the 30° angle is 6 cm, what is the length of the hypotenuse?