In a certain isosceles right triangle, the altitude to the hypotenuse has length $6$. What is the area of the triangle?
1. Input data entered: angle α, angle β, angle γ and height h
alpha = 45° ; ; beta = 45° ; ; gamma = 90° ; ; h = 6 ;
2. From height h and angle α we calculate side a:
a = { h }/{ sin beta } = { 6 }/{ sin (45° ) } = 8.485 ; ; b = { h }/{ sin alpha } = { 6 }/{ sin (45° ) } = 8.485 ;
3. From side a and angle α we calculate hypotenuse c:
sin alpha = a:c ; c = a/ sin alpha = 8.485/ sin(45 ° ) = 12 ; ;
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.
a = 8.49 ; ; b = 8.49 ; ; c = 12 ; ;
4. The triangle area - from two legs
Area = { ab }/{ 2 } = { 8.49 * 8.49 }/{ 2 } = 36 Sq. units.
