How do you find the height of a right-angled triangle?

Question

I need to know how to find the height a of right angled triangle without measuring it if I know the lengths of the sides. This is so I can work out the area, which I already know how to do, but in some of the problems the height of the triangle isn't given, only the side measurments are.
E.g a triangle with sides of 50mm, 13mm and 12mm. How would I work this out? Please explain!!!

Answer 1

To find the area of a triangle which is not right-angled, when you know only the sides, there are two techniques you can use.

Let ABC be the triangle, with sides of length a, b and c opposite angles A, B and C respectively. Drop a perpendicular from B to meet CA at D. Let the length of this perpendicular be h.

Method 1:

Then in thriangle ABD:
h / c = sin(A)
h = c sin(A) ................(1)

The area S of ABC is:
S = bh / 2 ..........(2)
Substituting for h from (1) in (2) gives an alternative formula for the area:
S = bc sin(A) / 2 ..............(3)

As you still need to know sin(A), you can use the cosine rule to find angle A as follows:
cos(A) = ( b^2 + c^2 - a^2 ) / 2bc.

Once you know cos(A), you can either find A and then its sine, using a calculator, or alternatively use the formula:
sin^2(A) = 1 - cos^2(A).

Once you know sin(A), you can use (1) above to find h or (2) above to find the area directly.

Method 2:

There is a more advanced formula for finding the area of the triangle directly from its sides. That is:
S = sqrt( s(s - a)(s - b)(s - c) ), where s = (a + b + c) / 2.

Answer 2

OK. U know that area of triangle ==1/2 base x height. Longest side=50mm. Picture(or draw) triangle with 2 shorter sides as base & height. Doesn't matter which. Then multiply to find area. Base =12mm Ht.=13. Area= 6mm (1/2 base) x 13mm(ht) = 78mm2. OR base=13mm, ht=12mm. 1/2 base=6.5mm ht=12mm. Area= 6.5mm x 12mm = 78mm2. Hope this helps.

Answer 3

In a right triangle, there is no necessity to consider the height. To determine its area simply multiply the lengths of the two sides that touch the right angle and divide by two. The reason for this is that it is half of a rectangle.

Answer 4

use the toa-soh-cah formula. http://mathworld.wolfram.com/Trigonometry.html