On a right triangle, a² + b² = c², and the two special triangles have fixed side ratios you should not re-derive. The ACT does not give you these. They belong on the list in ACT Math formulas.
The theorem
c is the hypotenuse, the side opposite the right angle. The legs are a and b. If the legs are 5 and 12, the hypotenuse is 13. If you know the hypotenuse and one leg, subtract the square of the leg from the square of the hypotenuse, then take the square root. This is also how coordinate distance works, covered in midpoint and distance.
45-45-90 and 30-60-90
A 45-45-90 triangle has legs x and a hypotenuse x√2. A 30-60-90 triangle has sides x, x√3, and 2x. The shortest side is opposite 30°. The longest is opposite 90°. If a question says “isosceles right triangle,” use 45-45-90 and stop calculating from scratch.
Trig ratios for the same triangles are in right-triangle trigonometry. The plane those points sit on is in coordinate geometry.
The bottom line
Square the legs and add when the triangle is ordinary. Use the side ratios when the angles are 45-45-90 or 30-60-90.
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