Derive and use trig ratios for 30-60-90 and 45-45-90 special right triangles.
Apply the correct special right-triangle family in context
Problem
A ramp makes a \(30°\) angle with the ground and reaches a vertical height of \(6~\text{ft}\). Treat \(6~\text{ft}\) as the short leg of a \(30-60-90\) triangle and calculate the ramp length and horizontal run exactly.
The ramp angle identifies the vertical rise as the side opposite thirty degrees, so it is the short leg. We’ll match the context to the thirty-sixty-ninety family, scale the exact ratio, assign the hypotenuse to the ramp and long leg to the run, and verify the dimensions by squaring.
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