CPhill

Assuming an isosceles trapezoid

80 = (1/2) h  (b + b + 12)

160 = h ( 12 + 2b)

80 = h ( 6 + b)

80 / ( 6 + b)  = h

We have a right triangle

One leg = h

The hypotenuse = 14 

One leg =  b + 12 - 6  =  b + 6

So

[ 80 / (b + 6)]^2  + (b + 6)^2  = 14^2    let (b + 6)^2  =  x

6400/x  + x   = 14^2         multiply through by x  

6400 + x^2 = 196x

x^2 -196x + 6400  = 0

x ≈ 154.6 or x = 41.936

(b + 6 )^2 =  154.6   → b ≈ 6.43

(b + 6)^2 = 41.936  → b ≈ .476

So

The median =   ( b + b + 12) / 2 =  b + 6  =  either ≈  b + 6 ≈  12.43  or ≈  6.476

Let the total that he started with = T

In the  first store he spends   T/2 + 10

What he has left =   T - (T/2 + 10)  =  T/2 -10

In the second store he spends   (1/2) (T/2 -10) + 10  =  T/4  + 5

What he has left  =  (T/2 - 10) - ( T/4 + 5)  = T/4 -15

In the third store he spends    (.80) (T/4  -15) =  (4/5)(T/4 - 15) = T/5 -12

What the has left  =  (T/4 -15) - (T/5 -12)  =  T/20 -3

And what he has left = $5

So

T/20 - 3 = 5

T/20  = 8

T = 20 *8  = $160  =  what he started with

Law of Sines

sin 70 / 25 = sin C / 26

sin C =  26 * sin 70 / 25

arcsin [ 26 * sin 70 / 25 ]  = C  ≈  77.8°

Draw a unit circle at the origin, O, and let P and Q loe pn this  circle

tan OP  = 4/1

cos OP = 1/sqrt(4^2 + 1^2)  = 1/sqrt 17      sin OP  = 4/sqrt 17

tan  OQ  = 5/1

cos OQ = 1/sqrt (5^2 + 1^2)  = 1/sqrt 26     sin QP = 5/sqrt  26

We can write P  as ( 1/sqrt 17 , 4/sqrt 17)

We can write Q as ( 1/sqrt 26 , 5/sqrt 26)

Slope  PQ  =

(5/sqrt 26 - 4/sqrt 17)  / ( 1/sqrt 26 - 1/sqrt 17)  =

(5sqrt 17- 4sqrt 26) / ( sqrt 17 - sqrt 26)  =

( 5 sqrt17 -4sqrt 26) (sqrt 17 + sqrt 26) / (17 -26)  =

(5*17 + sqrt 26 * sqrt 17 - 4*26) / -9

(-19 + sqrt 442) / -9 = 

(sqrt 442 - 19) / 9   ≈  -0.224

AB = 4  ???

https://web2.0calc.com/questions/geometry_50437

69 / 2553 = x / 1110

1110 * 69  = 2553 * x

x = 1110 * 69 / 2553   =  30

https://web2.0calc.com/questions/geometry_38859

Shortest altitude is drawn to the longest side

Area  =  sqrt [ 17 * (17 - 15) *(17 -10) * (17 - 9) ] = sqrt [17 * 2 * 7 * 8] = 4sqrt (119)

4sqrt (119)  = (1/2) 15 * shortest altitude

(8/15)sqrt (119) = shortest altitude ≈  5.82

See a similar problem here : 

https://web2.0calc.com/questions/trapezoid_85