SOLUTION: 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

Algebra ->  Triangles -> SOLUTION: 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      Log On


   

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?
Answer by jim_thompson5910(35256) 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=%281%2F2%29%2Ab%2Ah


A=%281%2F2%29%2A14%2A12 Plug in b=14 and h=12


A=168%2F2 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=%281%2F2%29%2Ab%2Ah Go back to the area of a triangle formula


84=%281%2F2%29%2A13%2Ah Plug in A=84 and b=13


84=%2813%2Ah%29%2F%282%29 Multiply.


168=13%2Ah 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=%281%2F2%29%2Ab%2Ah Go back to the area of a triangle formula


84=%281%2F2%29%2A15%2Ah Plug in A=84 and b=15


84=%2815%2Ah%29%2F%282%29 Multiply.


168=15%2Ah 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