Finding the Hypotenuse

This lesson covers the most common Pythagoras question: both legs known, hypotenuse missing. Legs of 9 and 12 give 81 plus 144, which is 225, so c is the square root of 225, which is 15. The rule to remember is that finding the hypotenuse means adding — you know the two small squares and you are building the big one, so the areas add. Then sense-check: 15 is bigger than the longest leg of 12, and smaller than 9 plus 12, which is 21. Both conditions hold. If your answer had come out under 12, you would know immediately that something went wrong.

Most hypotenuses are not whole numbers. Legs of 5 and 9 give 25 plus 81, which is 106, so c is the square root of 106 — about 10.30 centimetres to two decimal places. Read the question to see which form it wants. If it says 'exact answer' or 'in surd form', leave it as root 106, because rounding would actually lose marks. If it says 'to one decimal place' or 'to the nearest centimetre', round it and keep the units. Sense-check as always: 100 is less than 106 which is less than 121, so the answer sits between 10 and 11 — and 10.30 does.

Rectangles are the most common disguise for a Pythagoras question, because every corner is already a right angle. Draw the diagonal and it splits the rectangle into two right-angled triangles, with the diagonal as the hypotenuse and the two sides as the legs. For a screen 48 by 20: 2304 plus 400 is 2704, and the square root of 2704 is 52 centimetres. That is why a television is sold as a '50 inch' set — the number quoted is the diagonal, not the width. Watch for the same shape in gates with a diagonal brace, a suitcase that has to fit a rectangular opening, or a diagonal path across a rectangular field.

Word problems hide the right angle in the wording. Someone walks 300 metres east, then 400 metres north — east and north are perpendicular, so that is a right-angled triangle. 90000 plus 160,000 is 250,000, and the square root of that is 500 metres. Notice they walked 700 metres in total but ended up only 500 metres from the start. Phrases that signal a right angle include 'due north' and 'due east', a ladder leaning against a vertical wall on level ground, and the height of a flagpole, since poles stand upright. Whatever the wording, always draw the triangle and label it first — a sketch turns a confusing paragraph into a diagram you have already solved.

Often Pythagoras is only the first step. Asked for the perimeter of a right-angled triangle with legs 6 and 8: step one finds the hypotenuse, 36 plus 64 is 100 so c is 10. Step two adds all three sides: 6 plus 8 plus 10 is 24 centimetres. Step three is to check you have answered what was actually asked — the perimeter, which is 24. Stopping at c equals 10 answers a question nobody asked, and it is a genuinely common way to lose marks. The same pattern shows up with the area of a right-angled triangle, the cost of fencing, the length of trim, and 'how much further'. Re-read the question after you find c.