lesson
Identify the hypotenuse first
1 · Learn
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other sides: c² = a² + b². The hypotenuse is opposite the right angle and is the longest side.
To find a shorter side, subtract the other square from the hypotenuse square. Do not apply this relationship to a triangle without evidence that it is right-angled.
Why does Pythagoras work? Arrange FOUR congruent right triangles, each with perpendicular sides a and b and hypotenuse c, inside a square of side a+b as shown. The central gap has four sides of length c. At each gap corner, the two acute triangle angles sum to 90°, leaving a right angle in the gap. Thus the gap is a square, not merely a rhombus.
Compare areas: the outer square is (a+b)² = a²+2ab+b². The four triangles occupy 4×ab/2 = 2ab. Subtracting them leaves the central square: c² = a²+b². The argument uses arbitrary positive a and b, not just one measured example. In the drawing, a=6 and b=8: outer area 196 minus four areas of 24 leaves 100, so c=10. All these areas are in square centimetres.
2 · Worked example
With perpendicular sides 6 cm and 8 cm, c² = 36 + 64 = 100, so c = 10 cm. The positive length is used.
3 · Your turn
A right triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the other side.
Check your answer
Its square is 169 - 25 = 144, so its length is 12 cm.
4 · Apply your learning
Use the supplied 6-8-10 right-triangle diagram. Identify the hypotenuse and check the squared-length equation. Cover one length at a time, recover it from the other two, and explain when to add squares and when to subtract them.
Adult guidance and safety
Ask for the reasoning and a check, not only a numerical result. Supply accurate diagrams, coordinate grids and data displays where required. Use varied examples and revisit prerequisite gaps; these short teaching models are not a completed assessment programme.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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