CPhill

Thanks for the correction, EP!!!!

We can use the Law of Cosines twice to find BC

Let the intersection of the median drawn from A to BC =  D

The  angles formed by the intersection of  the median with BC  will be supplementary...so, their cosines will have opposite signs.....so cos BDA  =  - cos CDA  .....so  -cos BDA  = cos CDA

So we have that

3^2 = [ (1/2) BC]^2 + (BC)^2  - 2[ (1/2)BC] [BC] cos BDA    (1)

4^2  = [ (1/2)BC]^2 + (BC)^2  - 2[(1/2)BC] [BC] cos CDA

4^2 = [ (1/2)BC]^2  + (BC)^2  - 2[ (1/2)BC][BC] (-cos BDA)       

4^2  = [ (1/2)BC]^2  + (BC)^2  + 2[ (1/2)BC][BC] (cos BDA)    (2)

Add (1) and (2)   and we get that

25  =  2[ (1/2)BC]^2 + 2[BC]^2      simplify

25  = 2 (1/4)BC^2 + 2BC^2

25  = (1/2)BC^2 + 2BC^2

25  = (2.5)BC^2        divide both sides by 2.5

10  = BC^2

BC  = √10

Here's a pic :

y = (1/2)x  + 3

y = -2x + 6

y = 1

See the graph, here : https://www.desmos.com/calculator/6vqbm9hnpq

The vertex points of the triangular region  are   ( -4, 1)   (2.5, 1)  and  (1.2, 3.6)

The base of the triangular region has a length of  ( 2.5 - - 4)  = 6.5

The height of the triangular region is  ( 3.6 - 1)  = 2.6

So.....the area of the region is

(1/2) ( 6.5) ( 2.6)   =  8.45  units^2

CORRECTED ANSWER

2)   Any value of x  that makes  25 - 2x  ≥ 0   but  NOT a perfect square  will mean that  a radical will still remain in the expression......and  the remaining radical WILL NOT be a rational number....to see this

Let x  =  2     and we have that

√[25 - (2)(2) ]  / 2  =  √21 / 2       and  √21/2  is NOT RATIONAL

So......we only get a rational  whenever x  = 8   or  x  = 12   thusly

√[25 - 2(8) ] / 2  =  √9 / 2  =   3/2    which  IS rational

And

√ [25 - 2(12) ] / 2  =  √1 /2  =    1/2  which is also rational

So...remember.....any positive integer  under the radical that is not a perfect square will produce an irrational

Hope that helps  !!!

1)    We cannot take the square root of a negative number and get a real number returned

Therfore.....the quantity under the radical MUST BE ≥ 0

So....

25  - 2x ≥  0        add 2x to both sides

25 ≥ 2x              divide both sides by 2

25/2 ≥  x       which is the same as   x ≤ 25/2

The limit of what  ???

On the left.....

The first graph here : https://prnt.sc/phm00h

The graph on the left  here : https://prnt.sc/phlsoe

No prob  !!!!