SOLUTION: Triangle ABC is inscribed in a semicircle of diameter a = 12 (a) If x denotes the length of side AC, express the length y of side BC as a function of x. (Hint: Angle ACB is a ri

Algebra ->  Functions -> SOLUTION: Triangle ABC is inscribed in a semicircle of diameter a = 12 (a) If x denotes the length of side AC, express the length y of side BC as a function of x. (Hint: Angle ACB is a ri      Log On


   

Question 1145973: Triangle ABC is inscribed in a semicircle of diameter a = 12
(a) If x denotes the length of side AC, express the length y of side BC as a function of x. (Hint: Angle ACB is a right angle.)
(b) Express the area 𝒜 of triangle ABC as a function of x.
(c) State the domain of this function.
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.

We have right angled triangle ABC with the right angle at vertex A.

 
The hypotenuse length is "a" units, and the leg AC has the length x (given).



(a)  Apply Pythagorean theorem :  | BC | = sqrt%28a%5E2-x%5E2%29 = sqrt%2812%5E2-x%5E2%29.


     it is the expression you are looking for.




(b)  Area is half of the product of leg lengths


        A = %281%2F2%29%2Ax%2Asqrt%2812%5E2-x%5E2%29.



(c)  The domain of this function is  0 < x < 12.


      The endpoints 0 and 12 are not included to the domain, since otherwise the triangle becomes degenerated.