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# Altitudes

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How many units are in the sum of the lengths of the three altitudes in a triangle with sides  5, 12, and 13?

Jun 19, 2022

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Note that we have a right triangle, because $$5^2 + 12^2 = 13^2$$

This means that 2 of the altitudes are 5 and 12 because they are perpendicular to another side.

The third altitude can be found with this formula: $$\text{hypotenuse} \times \text{altitude} \div 2 = \text{Area}$$

We know the area is 30 and the hypotenuse is 13, meaning the altitude is $${60 \over 13}$$

Can you find the sum from here?

Jun 19, 2022