SOLUTION: How do I find the height of an scalene triangle, where not angle degrees are provided, only the following side dimensions?: The base=14cm, the longest side=15cm and the shortest si

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Question 975346: How do I find the height of an scalene triangle, where not angle degrees are provided, only the following side dimensions?: The base=14cm, the longest side=15cm and the shortest side=13cm
Found 2 solutions by josgarithmetic, solver91311:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Pick either end of the base for dealing with the angle there, and maybe use Law of Cosines. Choosing the sides 13 and 14,
13%5E2%2B14%5E2-2%2A13%2A14%2Acos%28x%29=15%5E2

13%5E2%2B14%5E2-15%5E2=2%2A13%2A14%2Acos%28x%29
cos%28x%29=%2813%5E2%2B14%5E2-15%5E2%29%2F%282%2A13%2A14%29.

What you want is height, which is highlight%2813%2Asin%28x%29%29. Compute cosine, and use the identify sin%5E2%28x%29%2Bcos%5E2%28x%29=1 to find sin(x). You should be able to finish this.


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cos%28x%29=5%2F13.
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sin%5E2%28x%29=1%5E2-%285%2F13%29%5E2
sin%5E2%28x%29=%28169-25%29%2F169
sin%5E2%28x%29=144%2F169
sin%28x%29=12%2F13
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Height is 13%2812%2F13%29=highlight%2812%29.

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


The formula for the height of a scalene triangle is dependent on the measure of the base to which you want to measure the height (which you did not specify) and the area of the triangle.

So the first thing we are faced with is the problem of finding the area of the triangle knowing the measures of the three sides. Fortunately, Pythagoras wasn't the only old Greek guy who knew something about Geometry. Heron (or Hero) of Alexandria left us with the following formula:



Ok, , , and are the measures of the sides, but what is ?

Actually, is simply the semi-perimeter, which is to say the perimeter divided by 2. In the case of your triangle, and half of that is . Now we can calculate the area.



I'll leave that arithmetic up to you.

Once you have the area, and have decided which of the three sides you want to consider the base, you can calculate the height with respect to that base using:



John

My calculator said it, I believe it, that settles it