<h3>Pythagoras' theorem</h3><p>In a right-angled triangle, the side opposite the right angle is the <strong>hypotenuse</strong>. Pythagoras' theorem states <code>a² + b² = c²</code>, where c is the hypotenuse. If the two shorter sides are 3 and 4, then <code>c² = 9 + 16 = 25</code>, so <code>c = 5</code>.</p><h3>Trigonometric ratios</h3><p>For an acute angle in a right-angled triangle, remember <strong>SOH-CAH-TOA</strong>:</p><ul><li><code>sin θ = opposite ÷ hypotenuse</code></li><li><code>cos θ = adjacent ÷ hypotenuse</code></li><li><code>tan θ = opposite ÷ adjacent</code></li></ul><div class="callout"><strong>Kenyan context:</strong> A surveyor stands 40 m from the foot of a mobile-network mast in Thika and measures the angle to the top as 35 degrees. The height is <code>40 x tan 35° ≈ 40 x 0.700 = 28</code> metres. A ladder leaning so its foot is 1.5 m from a wall and its top 2 m up reaches a length of <code>√(1.5² + 2²) = √6.25 = 2.5</code> m.</div>
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In a right triangle with shorter sides 3 and 4, the hypotenuse is: