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neurologyYou need to know

• Pythagoras’ Theorem applies only to right-angled triangles (triangles with one `90^\circ` angle).
• In a right-angled triangle, the hypotenuse is the side opposite the right angle and it is always the longest side.
• If the shorter sides are `a` and `b` and the hypotenuse is `c`, then Pythagoras’ Theorem is `a^2 + b^2 = c^2`.
• To find the hypotenuse, you add the squares of the two shorter sides and then take the square root: `c = \sqrt{a^2 + b^2}`.
• To find a missing shorter side, you subtract the known shorter side squared from the hypotenuse squared and then take the square root: `a = \sqrt{c^2 - b^2}` (or `b = \sqrt{c^2 - a^2}`).
• The calculation `c^2 - b^2` (or `c^2 - a^2`) must be positive; otherwise the given lengths cannot form a right-angled triangle with `c` as the hypotenuse.
• Pythagoras’ Theorem and its rearrangements involve squaring (multiplying a number by itself) and square rooting (finding a number that squares to give the original).
• When checking if a triangle is right angled, you must square all three side lengths and compare the largest side squared with the sum of the other two squared.
• A triangle is right angled if (and only if) the squares satisfy `a^2 + b^2 = c^2` where `c` is the longest side (the converse of Pythagoras’ Theorem).
• If `a^2 + b^2 > c^2`, the triangle is acute; if `a^2 + b^2 < c^2`, the triangle is obtuse (with `c` the longest side).
• Side lengths must be in the same units before using Pythagoras’ Theorem.
• If a length is not an integer, it may be left in surd form (for example `\sqrt{13}`) or approximated using a decimal.
• When rounding a decimal length, the degree of accuracy (for example, 1 decimal place) should be stated and used consistently.
• In a diagram, the right angle is often marked with a small square; the hypotenuse is the side opposite this mark.
• Pythagoras’ Theorem gives a unique positive length because side lengths are positive (you take the positive square root).

• Identify the hypotenuse correctly by locating the right angle and selecting the side opposite it.
• Substitute given side lengths into `a^2 + b^2 = c^2` using consistent units, and calculate squares accurately.
• Rearrange the theorem to make a shorter side the subject (for example `a = \sqrt{c^2 - b^2}`) and evaluate it, taking the positive square root.
• Check that the hypotenuse used is the longest side before subtracting to find a shorter side, to avoid impossible calculations.
• Solve Pythagoras problems from a labelled diagram by setting up a clear equation first, then showing ordered working to the final length.
• Determine whether a triangle is right angled by ordering the sides, squaring them, and testing whether `a^2 + b^2` equals `c^2` exactly.
• Use the result of the comparison (`=`, `>`, or `<`) to classify a triangle as right, acute, or obtuse, stating the conclusion clearly.
• Round or leave answers appropriately (decimal or surd), and include correct units in the final statement.