Questions on Geometry: Triangles answered by real tutors!

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Question 148446: 10. If the sides of a triangle are 13, 14, and 15 cm long, then the altitude drawn to the 14-cm side is 12 cm long. How long are the other two altitudes? Which side has the longest altitude?
: 10. If the sides of a triangle are 13, 14, and 15 cm long, then the altitude drawn to the 14-cm side is 12 cm long. How long are the other two altitudes? Which side has the longest altitude?

Answer by jim_thompson5910(9217) About Me  (Show Source):
You can put this solution on YOUR website!
First let's draw the picture.

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From the drawing, we can see that the base is 14 cm and the height is 12 cm.

Note: the "base" can be any of the three sides.

Remember, the formula for the area of any triangle is

A=(1/2)*b*h


A=(1/2)*14*12 Plug in b=14 and h=12


A=168/2 Multiply

A=84 Reduce

So the area of the triangle is 84 square cm


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Now, if we make the base any other side (let's say the side labelled "13 cm"), then the base becomes b=13

A=(1/2)*b*h Go back to the area of a triangle formula


84=(1/2)*13*h Plug in A=84 and b=13


84=(13*h)/(2) Multiply.


168=13*h Multiply both sides by 2.


12.92308=h Divide both sides by 13 to isolate "h".


So when the base is 13 cm, the height is 12.92308 cm



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Finally, if we make the base the last side, then the base is b=15

A=(1/2)*b*h Go back to the area of a triangle formula


84=(1/2)*15*h Plug in A=84 and b=15


84=(15*h)/(2) Multiply.


168=15*h Multiply both sides by 2.


11.2=h Divide both sides by 15 to isolate "h".


So when the base is 15 cm, the height is 11.2 cm


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So the largest height is 12.92308 which occurs when the base is 13 cm