The sides of a triangle are 5, 12, and 13. What is the length of its shortest altitude?

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Accepted Answer

W e space k n o w space t h a t space t h e space a r e a space o f space t h e space t r i a n g l e equals 1 half cross times b a s e cross times h e i g h t G i v e n space s i d e s space o f space t h e space t r i a n g l e space a r e colon space 5 comma space 12 space a n d space 13 T h u s space t h e space g i v e n space s i d e s space a r e space P y t h a g o r e a n space t r i p l e t s space a n d space h e n c e space i t space i s space r i g h t space t r i a n g l e. T h e r e f o r e comma space b a s e equals 12 comma space h e i g h t equals 5 space a n d space h y p o t e n u s e equals 13 A r e a equals 1 half cross times 12 cross times 5 space space space space space space space space equals 30 space s q. u n i t s N o w space c o n s i d e r space h y p o t e n u s e space a s space i t s space b a s e. space L e t space apostrophe h apostrophe space b e space t h e space a l t i t u d e space f r o m space t h e space r i g h t space a n g l e d space v e r t e x space t o space t h e space h y p o t e n u s e. A r e a equals 1 half cross times 13 cross times h rightwards double arrow 30 equals 1 half cross times 13 cross times h rightwards double arrow h equals fraction numerator 30 cross times 2 over denominator 13 end fraction rightwards double arrow h almost equal to 4.62 space u n i t s