A triangle has sides of lengths 5 cm, 12 cm, and 13 cm. If a perpendicular is drawn from the vertex opposite the longest side to the longest side, what is the length of the perpendicular? (Rounded off to two decimal places)
4.62 cm
Since \(5^2+12^2 = 25+144 = 169 = 13^2\), the triangle is right-angled with the right angle between the sides 5 cm and 12 cm.
Area of the triangle: \(\tfrac12\times5\times12 = 30\) sq cm.
The perpendicular from the right-angle vertex to the hypotenuse (the longest side, 13 cm): \(h = \dfrac{2\times\text{Area}}{13} = \dfrac{60}{13} \approx 4.62\) cm.
Hence, the length of the perpendicular is approximately 4.62 cm.
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