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a - b = (-1 -3 , -4 -2) = (-4, -6)
https://web2.0calc.com/questions/geometry_7281
https://web2.0calc.com/questions/geometry_63597
https://web2.0calc.com/questions/trapezoid_139
A
8
B 2 K 3 C
Because AK is an altitude
AK = sqrt [ AC^2 - CK^2 ] = sqrt [ 8^2 - 3^2 ] = sqrt [ 55 ]
AB = sqrt [ AK^2 + BK^2 ] = sqrt [ 55 + 4 ] = sqrt [ 59 ]
(r-5)^2 + (r- 5)^2 = r^2
2 [ r^2 - 10r + 25 ] = r^2
2r^2 - 20r + 50 = r^2
r^2 - 20r = - 50 complete the square on r
r^2 - 20r + 100 = -50 +100
(r - 10^2 = 50 take the positive root
r - 10 = sqrt 50
r -10 = 5sqrt 2
r = 5sqrt 2 + 10 ≈ 17.07
G
2 8
F E
FE = sqrt [ 8^2 - 2^2] = sqrt 60 = 2 sqrt 15
sin G = FE / EG = 2sqrt (15) / 8 = sqrt (15) / 4
sec G = GE / FG = 8 / 2 = 4
tan E = FG / FE = 2 / [ 2sqrt 15] = 1 /sqrt 15 = sqrt (15) / 15
N
C M B
AC^2 + (BC/2)^2 = 1
(AC/2)^2 + BC^2 = 1 simplify this system
AC^2 + BC^2 / 4 = 1
AC^2/4 + BC^2 = 1
4AC^2 + BC^2 = 4
-AC^2/4 - BC^2 = -1 add these
(15/4)AC^2 = 3
AC^2 = 4/5
And
4/5 + BC^2 / 4 = 1
BC^2 / 4 = 1/5
BC^2 = 4/5
Hypotenuse length = sqrt [ AC^2 + BC^2 ] = sqrt [ 4/5 + 4/5 ] = sqrt (8/5) = 2sqrt (2) / sqrt (5) =
2 sqrt (10) / 5
British Flag Theorem
AP^2 + CP^2 = BP^2 + DP^2
3^2 + 3^2 = 3^2 + DP^2
9 = DP^2
DP = 3
