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# Altitudes of a triangle

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

Apr 13, 2022

#1
+1

\(8+12=20\)

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

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   Apr 13, 2022
edited by CPhill  Apr 13, 2022
edited by CPhill  Apr 13, 2022
#3
+2

Sorry!

Its more complicated than I thought!!! Vinculum  Apr 13, 2022
#4
0

I think you might have assumed a right triangle with legs of 8 and 12....in that case, your answer would be correct    CPhill  Apr 13, 2022
#5
-3

Yep he sure would Chris

Apr 13, 2022
#6
-3

hey chris did you make a request for a modship for me

Apr 13, 2022
#7
+1

You should DM him...

BuilderBoi  Apr 13, 2022
#8
+4

Yeah man, this kind of stuff should be talked about privately......

Vinculum  Apr 13, 2022
#9
-2

Good point

Apr 13, 2022