ACT Preparation / Unit 13 ACT 1227 worked example, 60 to 90 seconds
Special Right Triangle Sides
The idea
Special right triangle problems are solved in three steps. First, identify the triangle from its angles or its shape: a square cut by its diagonal gives 45-45-90, and an equilateral triangle cut by its height gives 30-60-90. Second, find the base side: the leg for 45-45-90, the shorter leg for 30-60-90. Third, multiply by the ratio for each other side, and rationalize if √2 or √3 lands in a denominator.
In practice
Worked example: an equilateral triangle has sides of 12. Its height cuts it into two 30-60-90 triangles with hypotenuse 12 and shorter leg 6, so the height, the longer leg, is 6√3. Second: a 45-45-90 triangle has hypotenuse 9, so each leg is 9/√2 = 9√2/2.
Where people slip
Match each side to the angle it faces: the shorter leg is always opposite 30°, and the longer leg, √3 times it, is opposite 60°.
Your place on the trail
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