CPhill

pi/4

∫       [sec x(2tanx - 5secx) ] dx  =

0

pi/4                         pi/4

∫    sec x(2tanx)      -       ∫   5 sec^2x  dx  =

0                              0

  pi/4                                       pi/4

2 ∫    sec x tanx  dx     -       5  ∫   sec^2x   dx  =

  0                                            0

             pi/4                               pi/4

2 secx ]                   -      5 tan x ]          =

            0                                    0

2 [ √2 - 1 ]    -   5 [ 1 - 0 ]  =

2√2 - 2 - 5  =

2√2  - 7

However.....this represents an area  so the correct answer  is 

abs ( 2√2  - 7 )  ≈  4.1716  units^2

Drawing an altitude from the apex to the base forms two 30-60-90  right triangles

In one of the triangles, the base  = 10   and this is the side opposite the 60° angle

The height of this triangle [ and of the larger triangle] will be opposite the 30° angle and wil be 10/√3  units long

So....the area of the biggger triangle =  (1/2)base * height =

(1/2) 20 * 10/√3  =  100/√3    sq units

Here's a pic

Here's another way using just Geometry

Since   ACB is bisected......then  DC divides  AB   in a ratio of  12 /15  = BD/ DA

So.....BD  =  (12/27) * 9  =   (4/9)*9  = 4

Using the Pythagorean Theorem

CD  = √[ BD^2  + BC^2] =  √ [ 4^2  + 12^2 ]  =  √ [ 160 ]  ≈  12.65

I'm assuming that CD  is drawn to side AB

We have a Pythagorean right triangle  9 -12 - 15

Angle  ACB can be expressed  using the inverse tangent

arctangent (9/12)  =  ACB

And DCB =  .5arctangent (9/12)

Likewise, angle BDC = 90 - DCB = 90 - .5*arctangent (9/12)

So....using the Law of  Sines, we have that

CD/ sin DBC  =  BC /  sin DCB       →  DBC  = 90  →  sin 90  =  1

CD  =  12 / sin  ( 90 - .5 *arctangent (9/12) )   ≈  12.65

Here's a pic

9x-2y = -3,  - 3x+5y=27

Note that if we multiply the second equation by 3, we have

-9x + 15y = 81

So now we have

  9x  -  2y   = -3

- 9x + 15y = 81          add  these  and we have

13y  =  78             divide both sides by 13

y =  6

And to find x we can use

9x - 2(6)  = -3

9x -  12  = -3    add 12 to both sides

9x  =  9         divide both sides by 9

x =  1

So....the answer is (x,y) = (1, 6 )

Here's how the formula is derived :

Since the radius, r, of the incircle can be drawn to meet each side of the larger triangle at right angles, then we can construct three triangles by connecting each vertex of the larger triangle to its incenter.  And each of these triangles has a height of r and a side as a base.

So.....the total area, A, of the the larger triangle is

(1/2) r ( a + b + c)  =  A      where a,b,c are the side lengths

Multiply both sides by 2

r (a + b + c)  = 2A

Divide both sides by (a + b + c)

r =  2A / (a + b + c)

But (a + b + c)  is the perimeter = P

So....

r =  2A / P

281.25 / 45  = 6.25  =   625%

Note that

2.4^2  =  5.76 → √ 5.76 =  2.4

2.5^2  = 6.25   → √6.25  = 2.5

2.6^2  = 6.76   → √6.76  = 2.6

2.7^2  = 7.29  → √7.29  = 2.7

So

√ 5.76 + √6.76  =  2.4 + 2.6  = 5.0    (1)

And

√6.25 + √7.29  =  2.5 + 2.7  =  5.2    (2)

But

(1)  < √6 + √7 <  (2)  

So  √6 + √7  must lie between  5.0  and 5.2

Thus.....√6 + √7  must lie between  5 and 6

OK....post your questions

62.5  

Add a 0, move the decimal two places to the right and write a % sign  =

6250%