Pythagoras Theorem [Proof and Simple Applications with Converse]

233. The Parallelogram Law · diagonals squared sum equals sides squared sum

AC² + BD² always equals AB² + BC² + CD² + DA².

CAB = 500AB = 500BC = 300.2BC = 300.2CD = 500CD = 500DA = 300.2DA = 300.2AC = 700.1AC = 700.1BD = 436BD = 436AB² + BC² + CD² + DA² = 250000 + 90100 + 250000 + 90100 = 680200AB² + BC² + CD² + DA² = 250000 + 90100 + 250000 + 90100 = 680200AC² + BD² = 490100 + 190100 = 680200AC² + BD² = 490100 + 190100 = 680200ABD
Parallelogram law: the sum of the squares of the diagonals equals the sum of the squares of the four sides: AC² + BD² = AB² + BC² + CD² + DA². For a parallelogram, where opposite sides are equal, this becomes AC² + BD² = 2(AB² + BC²).

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Selina ICSE: Pythagoras Theorem [Proof and Simple Applications with Converse]

What this lesson covers

Try to break it

Drag A, B, or D to reshape the parallelogram into a rectangle, a rhombus, or a skewed shape. The two diagonals change length, but the sum of their squares (AC² + BD²) always equals 2·(AB² + AD²). Try to find a shape where the equation breaks — impossible. The parallelogram law holds everywhere.

How you build it

Construct a parallelogram with diagonals.

  • Point tool: mark point A.
  • Point tool: mark point B.
  • Point tool: mark point D, off the line AB.
  • Segment tool: join A to B.
  • Segment tool: join A to D.
  • Parallel tool: click B, then line AD — a line through B parallel to AD.
  • Parallel tool: click D, then line AB — a line through D parallel to AB.
  • Point tool: mark C where the two parallel lines cross — ABCD is the parallelogram.
  • Segment tool: join A to C — the first diagonal.
  • Segment tool: join B to D — the second diagonal.

The proof, step by step

Prove that the sum of the squares of the diagonals of a parallelogram equals the sum of the squares of its sides.

  • Draw AN ⊥ CD (produced) and DM ⊥ AB. Since AB || CD, the perpendicular heights AN and DM are equal.
  • Apply Pythagoras in ΔANC: AC² = AN² + NC². Apply Pythagoras in ΔDMB: BD² = DM² + MB².
  • Observe that NC = CD + DN and MB = AB - AM. From congruent right triangles ADN and BCM, DN = CM and AN = DM.
  • Add the two equations. Substitute AN² = DM² and simplify. The cross terms cancel, leaving AC² + BD² = AB² + BC² + CD² + DA².

Worked example

In a parallelogram ABCD, AB = 10 cm and BC = 6 cm. If the length of diagonal BD is 8 cm, find the length of diagonal AC.

Using the parallelogram law: AC² + BD² = 2(AB² + BC²). Substituting values: AC² + 64 = 2(100 + 36) = 272. Thus, AC² = 208, so AC = √208 ≈ 14.42 cm.

  • 12 cm
  • 14.42 cm — correct
  • 15 cm
  • 16 cm
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