How do I figure out the height of a triangle?

The height of a triangle is defined as the perpendicular distance from an angle to a side. Now, since this is a right triangle, and depending on how it is situated, the height could be 10" or 12" if you observe from the two acute angles. But from the right angle, the height goes from the angle to AC (your hypotenuse). Finding this one will be tricky but can be done using the Pythagorean Theorem.

If you drop a height from the right angle to the hypotenuse, it breaks it into two parts. The length of one part is x (since we don't know and since the height doesn't perfectly bisect the hypotenuse), and the length of the other part is 15.62-x (since the total length is 15.62"). It really doesn't matter which side is x, but for my explanation, I'll use x as the side closer to the leg that is 10". This height creates two new right triangles, and you can use the Pythagorean Theorem to find out both the height and the missing side x.

One formula will end up being 10^2 = h^2 + x^2, and the other formula will end up being 12^2 = h^2 + (15.62-x)^2. You can solve the first equation to find that h^2 = 100 - x^2 and substitute that in to the second equation:

12^2 = h^2 + (15.62-x)^2
144 = 100 - x^2 + 243.9844 - 31.24x + x^2
144 = 343.9844 - 31.24x
-199.9844 = -31.24x
x = 6.4 (rounded)

Knowing this, we can find that missing height:
h^2 = 100 - x^2
h^2 = 100 - (6.4)^2
h^2 = 100 - 40.98
h^2 = 59.02
h = 7.68"

So it all depends on the specific height you're looking for - it's either 10", 12", or 7.68".

As for your trough, surface area is the sum of all the areas. Break your trough into parts that you can find areas of. If you're just looking for the outer surface area, for example, you have two triangles for the ends and two rectangles for the sides (assuming that it's an open trough). The area of the triangles is A = 0.5bh, and the area of the rectangle sides is A = bh (b in this case will be a side of the trianlge, 2, and h will be the length of the trough, 16). So just add up all the surfaces that will comprise your trough and you should have your surface area. (I'm not sure if you need to include the inside as well, but if so you need 2 triangles and 2 rectangles for the outside, and 2 triangles and 2 rectangles for the inside as well.)

Hey thanks... I forgot about the Pythagorean theory... LOL and I'm trying to figure out the missing side thinking its impossible, and as soon as I read Pythagorean theory on your answer I knew.

Thanks a lot T.

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