Explanation:
I am not sure about the units in 11. to 14.
is it also m (meter) ? the please add this to the result numbers.
11.
Pythagoras at the core :
x² + y² = (22×sqrt(3))² = 22²×3 = 484×3 = 1452
then we can use the law of sine :
a/sin(A) = b/sin(B) = c/sin(C)
with the sides being opposite of the related angles.
so,
22×sqrt(3)/sin(90) = y/sin(60)
22×sqrt(3)/1 = y/sin(60)
y = 22×sqrt(3)×sin(60) = 33
and now in the original Pythagoras equation :
x² + 33² = 1452
x² = 1452 - 33² = 1452 - 1089 = 363
x = sqrt(363) = 19.05255888... ≈ 19.1
12.
the same basic approach :
x² + sqrt(6)² = y²
x² + 6 = y²
then the law of sine :
y/sin(90) = y/1 = sqrt(6)/sin(30) = sqrt(6)/0.5
y = sqrt(6)/ 1/2 = sqrt(6)×2/1 =
= 2×sqrt(6) = 4.898979486... ≈ 4.9
back to the original Pythagoras :
x² + 6 = 4.9²
x² = 24 - 6 = 18
x = sqrt(18) = 4.242640687... ≈ 4.2
13.
the same basic approach again :
y² = x² + (4×sqrt(21))² = x² + 16×21 = x² + 336
the law of sine :
y/sin(90) = y/1 = 4×sqrt(21)/sin(60)
y = 21.16601049... ≈ 21.2
21.2² = x² + 336
448 - 336 = x² = 112
x = sqrt(112) = 10.58300524... ≈ 10.6
14.
there is a line shared by both triangles.
let's call it "a".
y² = z² + a²
a/sin(90) = a/1 = 17/sin(30) = 17/0.5 = 2×17 = 34
a² = 17² + x²
34² = 17² + x²
1156 = 289 + x²
x² = 1156 - 289 = 867
x = sqrt(867) = 29.44486373... ≈ 29.4
y/sin(90) = y/1 = a/sin(45) = 34/sin(45)
y = 48.08326112... ≈ 48.1
48.1² = z² + 34²
2312 = z² + 1156
z² = 2312 - 1156 = 1156
z = 34
15.
equilateral means all sides are equally long.
since the base is 8 m, it means that also both roof sides are 8 m.
the height in such a triangle creates a right-angled triangle with one roof side as Hypotenuse and half the base as second leg.
8² = height² + (8/2)² = height² + 4²
64 = height² + 16
height² = 64 - 16 = 48
height = sqrt(48) = 6.92820323... ≈ 6.9 m
16.
the same as in 15. just with side lengths 10 cm.
10² = height² + 5²
100 = height² + 25
height² = 100 - 25 = 75
height = sqrt(75) = 8.660254038... ≈ 8.7 cm
17.
the diagonal of a rectangle is the Hypotenuse of a right-angled triangle with one length and one width of the rectangle as legs.
diagonal² = 48² + 14² = 2304 + 196 = 2500
diagonal = 50 ft
so, the rope must be at least 50 ft long.
18.
remember, the sum of all angles in a triangle is always 180°.
the standing person and his shadow enclose a 90° angle.
the sun rays make an 60° angle with the ground, so the angle at the head of the standing person is
180 - 90 - 60 = 30°
and now remember the law of sine as above.
the standing person is opposite of the 60° angle, the shadow in the ground is posit of the 30° angle at the head.
so,
180/sin(60) = shadow/sin(30) = shadow/0.5 = 2×shadow
shadow = 180 / (2×sin(60)) = 90/sin(60) =
= 103.9230485... ≈ 104 cm