How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides \(8\), \(12\), and \(17\)?

Guest Apr 13, 2022

#1**+1 **

\(8+12=20\)

So the sum of the lengths of the two longest altitudes in that triangle is \(20\)

Vinculum Apr 13, 2022

#2**+2 **

The semi-perimeter of this triangle = (8 + 12 + 17) /2 = 37/2 = 18.5

The area of this triangle = sqrt ( 18.5 * ( 18.5- 8) (18.5 -12) ( 18.5 - 17) ) =

sqrt [ 18.5 * 10.5 * 6.5 * 1.5 ] =

sqrt [ 1893. 93.75 ]

The longest two altitudes will be drawn to the shortest two sides

So, using the area formula for a triangle, we have

sqrt (1893.9375 ) = (1/2)(8) *altitude 1

sqrt(1893.9375) / 4 = altitude 1 (1)

And

sqrt (1893.9375 ) = (1/2)(12) * altitude 2

sqrt (1893.9375 ) / 6 = altitude 2 (2)

Adding (1) and (2) we get

sqrt (1893.9375) / 4 + sqrt(1893.9375) / 6 ≈ 18.133 units

CPhill Apr 13, 2022