Turn the Triangle, Keep the Roles

Use the named angles and endpoints rather than up, down, left or right.

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Find the square corner and the side across from it. That side is the hypotenuse. Choose the acute angle in the question; its opposite side does not touch that angle. Its adjacent leg touches the angle but is not the hypotenuse. Turning a drawing does not change any of these relationships. Changing the chosen angle exchanges the leg roles.

Worked example

In triangle PQR, Q is the right angle, PQ = 15, QR = 8 and PR = 17. For angle R, PR is the hypotenuse, PQ is opposite and QR is the adjacent leg. Turning the paper leaves those answers unchanged.

Turned triangle GHI has right angle H, GH 20 units, HI 21 units and GI 29 units. The chosen acute angle is I. The drawing is schematic. — Use GHI for all questions. Side roles follow the selected angle, not the page direction.

Would you believe it?

Doubling every side of a 5-12-13 triangle gives a similar 10-24-26 triangle. Corresponding angles stay the same even though all lengths change.

OpenStax: right-triangle ratios

Question 1 GHI is right angled at H. Which side is its hypotenuse? GH HI GI

Question 2 For angle I, which side is opposite? GH GI HI

Question 3 For angle I, which side is the adjacent leg? GI HI GH

Question 4 Switch the chosen angle from I to G. What is the adjacent leg now? HI GI GH

Question 5 Rotate the paper again without changing the chosen angle I. What happens to its opposite side? The horizontal side must always become adjacent GH remains opposite I GI becomes opposite I because it is longest

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